Conditional thinning and multiplicative statistics of Laguerre-type orthogonal polynomial ensembles

CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TISTICS OF LA GUERRE-TYPE OR THOGONAL POL YNOMIAL ENSEMBLES LESLIE MOLA G, GUILHERME L. F. SIL V A, AND LUN ZHANG Abstract. W e study the local statistics of orthogonal p olynomial ensem bles near a hard edge, sub ject to a m ultiplicative deformation of the measure. Probabilistically , this deformation corresp onds to a p osition-dependent conditional thinning of the particles. W e prov e that, under critical hard edge scaling and for a large class of p oten tials and deformation symbols, the correlation k ernel of the conditional ensem ble con verges to a univ ersal limit, whic h we identify as the conditional thinned Bessel point pro cess. W e derive an explicit expression for this limiting kernel in terms of the solution to a nonlo cal integrable system dep ending on a parameter. F or a sp ecial ch oice of the parameter, this system was recen tly iden tiﬁed in the study of multiplicativ e statistics of the Bessel p oint pro cess. Our results establish that this system gov erns the full correlation structure of the conditional Bessel point pro cess, extending the classical connection b et ween the standard Bessel kernel and the Painlev ´ e V equation. Contents 1. In tro duction 2 2. Statemen t of results 4 Organization of the paper 10 Ab out the notation 10 Ac kno wledgments 11 3. The model RHP 11 3.1. The model problem 12 3.2. Solv ability of the mo del problem 14 3.3. Diﬀeren tial equations for the mo del problem with some particular data 18 4. The model RHP: admissible data and asymptotics 23 4.1. The class of admissible h 24 4.2. Asymptotic analysis for Ψ ∞ as s → + ∞ 25 4.3. Asymptotic analysis of Ψ ∞ as u → + ∞ 29 4.4. Asymptotic analysis of the mo del RHP with admissible data 33 5. Consequences of the asymptotic analysis for the mo del problem 37 5.1. Con v ergence of the k ernel 37 5.2. Asymptotics for a relev ant in tegral 38 5.3. Pro of of main theorems on in tegrable equations 40 6. Asymptotic analysis of the RHP for OPs 42 6.1. Equilibrium measures and related quantities 42 6.2. The RHP for OPs 44 6.3. First transformation: in troduction of the g -function 45 6.4. Second transformation: opening of lenses 45 6.5. The global parametrix 46 6.6. The local parametrix near the hard edge 48 6.7. The local parametrix near the soft edge 51 6.8. Conclusion of the asymptotic analysis 52 7. Conclusion of main results 53 7.1. Asymptotics for the k ernel: pro of of Theorem 2.4 53 7.2. Asymptotics for the m ultiplicativ e statistics: proof of Theorem 2.6 55 App endix A. The Bessel parametrix 60 1 2 L. MOLAG, G. SIL V A, AND L. ZHANG App endix B. Asymptotic analysis for a class of integrals 63 References 64 1. Introduction The univ ersalit y of sp ectral statistics is a cornerstone of Random Matrix Theory (RMT). F rom its origins in nuclear ph ysics to mo dern applications in num b er theory , statistical mechanics, and high-dimensional data analysis, the lo cal correlation functions of eigenv alues ha v e prov en to b e robust in v ariants across a v ast arra y of mo dels, and a ric h ground for the prediction of new critical phenomena. Among the many striking connections b etw een random particle systems and other ﬁelds, the link to in tegrable systems has prov en exceptionally fruitful. This relationship is most celebrated at the “soft edge” of the sp ectrum, that is, close to the largest eigenv alue. The ﬂuctuations of the largest eigen v alues in Hermitian random matrix mo dels are described b y the Airy 2 p oin t pro cess, and the distribution of the largest particle is the T racy-Widom distribution F 2 . In turn, F 2 is go v erned b y the Hastings-McLeo d solution to the Painlev ´ e I I equation, a second order nonlinear integrable ODE. Suc h a connection betw een extremal eigen v alues and nonlinear diﬀeren tial equations w as pioneered b y T racy and Widom [ 52 ] and F orrester [ 36 ] for the Gaussian Unitary Ensem ble (GUE). These results w ere subsequen tly extended to a wide v ariet y of random matrix mo dels using diverse techniques [ 27 – 29 , 44 , 50 ]. But the connection b etw een strongly correlated particle systems and in tegrable systems extends well b ey ond RMT. The distribution F 2 describ es the ﬂuctuations of the longest increasing subsequences in random p ermutations [ 5 , 45 ] and, more generally , characterizes the univ ersal scaling limit of growth mo dels in the Kardar-P arisi-Zhang (KPZ) univ ersality class [ 3 , 48 ]. In recen t y ears, atten tion has expanded from standard gap probabilities to the study of m ultiplicativ e statistics of these p oin t pro cesses. It was discov ered that the generating functions for multiplicativ e statistics of the Airy 2 pro cess relate directly to the solution of the KPZ equation [ 3 , 9 , 17 , 33 , 48 ], and that the underlying p oin t pro cess, when weigh ted by these statistics, forms a new “conditional” ensem ble with a rich integrable structure [ 12 , 14 , 20 ]. F urthermore, this framework has b ecome a p o w erful to ol for extracting ﬁne-grained probabilistic data, such as large deviation estimates and tail asymptotics [ 12 , 14 , 24 , 55 ], while also motiv ating recent studies on higher-order soft edge scalings [ 10 , 11 , 15 , 16 , 21 , 26 , 41 ]. The developmen ts mentioned so far concern RMT ﬂuctuations around a so-called “soft edge”, and a parallel and equally ric h structure exists at the “hard edge” of the sp ectrum. A hard edge arises when eigenv alues are b ounded b y a strict physical constraint, most commonly the origin for p ositive deﬁnite matrices. The canonical mo del for this regime is the Wishart ensemble (or Laguerre Unitary Ensem ble - LUE), consisting of matrices of the form X X ∗ , where X is a rectangular matrix with complex Gaussian entries [ 58 ]. As the matrix size grows, the eigenv alues near the origin, when critically scaled, con v erge to the Bessel point process, a canonical p oin t pro cess whose correlations are describ ed by Bessel functions [ 36 , 39 ]. Fluctuations of eigen v alues near a hard edge also displa y a rich structure in connection with inte- grable systems. Historically , the gap probabilities for the Bessel pro cess w ere connected to the P ainlev ´ e V equation b y T racy and Widom [ 53 ], see also [ 43 ]. This connection was further extended to gener- ating functions and multi-in terv al statistic s ov er the years [ 4 , 19 , 25 ]. Ho w ev er, in light of the recent dev elopmen ts at the soft edge, it is natural to ask: T o what extent do es the ric h integrable structure of m ultiplicativ e statistics carry o v er at a hard edge? CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 3 In the present work, we provide an aﬃrmativ e answer to this question by explicitly constructing the k ernel of a limiting point pro cess associated with these m ultiplicativ e statistics, and describing it in terms of integrable systems. W e begin b y considering the eigenv alues of p ositive deﬁnite matrices of size n . W e sub ject these eigen v alues to a conditional thinning pro cedure, where eac h eigen v alue is retained or remov ed indep en- den tly with a position-dep endent probabilit y , and the ﬁnal process is the resulting ensemble conditioned that all particles ha ve b een retained. This pro cedure deﬁnes a ﬁnite- n conditional ensem ble. Our main result describes the asymptotic b eha vior, as n → ∞ , of the conditional ensemble in the critical regime where eigen v alues accum ulate near the hard edge. W e prov e that, under the hard-edge scaling, the correlation k ernel of the ﬁnite conditional ensem ble con v erges to a univ ersal limit. Our results are univ ersal, in the sense that w e establish them for a broad class of random matrix mo dels with a hard edge, as well as for a wide v ariet y of thinning proﬁles. W e iden tify this limit as the conditional thinned Bessel p oint pro cess, whic h corresp onds to applying the conditional thinning op eration directly to the classical Bessel p oint pro cess. F urthermore, our asymptotic analysis yields an explicit form ula for the correlation k ernel of this pro cess in terms of the solution to a nonlocal in tegrable system. The integrable structure of multiplicativ e statistics for the Bessel p oint pro cess was recen tly in v es- tigated by Ruzza [ 46 ], who show ed that they are gov erned by a sp eciﬁc nonlo cal integrable equation, whic h reco vers the classical Painlev ´ e V equation in the degenerate limit of gap probabilities. In our analysis, w e explicitly identify the nonlo cal system c haracterizing our limiting k ernel with the equation deriv ed by Ruzza. This identiﬁcation is crucial: it shows that the solution to this nonlo cal equation describ es not just the m ultiplicative statistics, but the full correlation k ernel of the conditional ensemble itself. Thus, the nonlo cal Painlev ´ e V transcenden t emerges as the fundamental structural inv arian t of the conditionally thinned Bessel pro cess. This result completes the description of conditional thinning statistics across the standard sp ectral regimes. The k ernel for the conditional thinning ensemble w as related to nonlo cal integrable systems at a regular soft edge in [ 37 ], and at a regular bulk p oint in [ 18 ] (see also [ 22 ] for further discussion, and [ 14 , 23 ] for related multiplicativ e statistics). Our work establishes the analogous theory for regular hard edges. Our analysis relies on the integrable structure of the ensembles via orthogonal p olynomials (OPs) and the asso ciated Riemann-Hilbert Problem (RHP) approach. The conditional thinned point process of an orthogonal polynomial ensem ble is itself an orthogonal p olynomial ensem ble, deﬁned b y a m ultiplicative deformation of the original w eigh t [ 20 ]. This observ ation serv es as the starting point for our RHP form ulation of the ﬁnite n correlation k ernel. While we apply the Deift-Zhou steep est descent metho d, our analysis deviates from the standard route in the construction of the local parametrix near the hard edge. Unlike the classical case, the jump matrices for this lo cal parametrix cannot b e reduced to piecewise constant matrices. Consequen tly , the resulting mo del problem inv olv es nontrivial p osition-dep endent jumps that enco de the symbol of the deformation. This model problem requires a separate asymptotic analysis, and ultimately it establishes the connection with the integrable systems that e merge. Finally , armed with the asymptotics for the conditional correlation kernel, w e recov er the multiplica- tiv e statistics using a recently established general deformation formula [ 38 ]. This approach b ypasses the traditional route using diﬀerential identities for Hank el determinan ts. 4 L. MOLAG, G. SIL V A, AND L. ZHANG 2. St a tement of resul ts The general family of mo dels we consider consists of deformations of the Laguerre-t yp e OP ensembles. More precisely , we consider random points x 1 , . . . , x n ∈ (0 , ∞ ) with joint distribution of the form 1 Z n ( s ) Y 1 ≤ i − 1 , x > 0 , (2.2) with σ n ( x ) = σ Q n ( x | s ) . . = 1 1 + e − s − n 2 m Q ( x ) , s ∈ R , m ∈ Z > 0 ﬁxed , (2.3) and the normalization constan t Z n ( s ) . . = Z ∞ 0 · · · Z ∞ 0 Y j 0 . (2.4) Our focus in the current pap er is in describing the eﬀect of the deformation σ n on eigen v alues near x = 0, that is, near the (regular) hard edge. As such, man y of our lo cal scaling limits will still hold if w e drop the regularit y condition on the soft edge x = a , or also if we drop the assumption that supp µ V is connected. Nev ertheless, we work under such conditions to simplify our analysis, as our main goal is not to work under the most general assumptions, but instead to compute the no v el scaling limits that arise at the hard edge, and sho w case their univ ersalit y . By ( 2.3 ), it is readily seen that σ n ( x ) → 1 as s → + ∞ , and ( 2.1 ) reduces to the classical Laguerre- t yp e OP ensem ble. F rom now on, whenev er we refer to the classical Laguerre-t yp e OP ensem ble, we CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 5 alw a ys mean ( 2.1 ) with the choice s = + ∞ , that is, σ n ≡ 1. With this degeneration in mind, the quotien t L Q n ( s ) . . = Z n ( s ) Z n (+ ∞ ) = E   n Y j =1 σ n ( x j | s )   (2.5) is a m ultiplicative statistic of the classical Laguerre-type OP ensemble, asso ciated to the symbol σ n ( x | s ). Alongside the in terest in the multiplicativ e statistics L Q n ( s ), the model ( 2.1 ) may also b e in terpreted as a deformed ensemble obtained when p erforming a conditional thinning pro cedure. Starting from the classical Laguerre-type OP ensemble with n particles x 1 , . . . , x n , each particle receiv es a mark 1 with probabilit y σ n ( x j ), and a mark 0 with complementary probability 1 − σ n ( x j ). The mark 1 is in terpreted as if the mark ed particle is “seen” in the system, and the mark 0 as if the particle is “lost” b y noise in the sample. This marking in terpretation corresp onds to the thinning step. Now construct a new particle system, the conditional thinned ensemble, obtained by conditioning that all the particles receiv ed the mark 1. A general framework dev elop ed by Claeys and Glesner [ 20 ] shows that ( 2.1 ) is precisely the densit y of particles in this conditional thinned ensem ble starting from the corresp onding Laguerre-t yp e OP ensemble. With the conditional thinned interpretation w e just explained in mind, it is natural to exp ect that statistics of ( 2.1 ) may dep end on prop erties of the function Q lab elling σ n . W e assume the function Q : [0 , ∞ ) → R satisﬁes the following conditions. Assumption 2.2. The function Q extends to an analytic function in a complex neighborho o d of [0 , ∞ ) suc h that Q ( x ) > 0 , x > 0 , and Q ( x ) = ( t x m (1 + O ( x )) , x → 0 , O ( x ϵ ) , x → ∞ , (2.6) for the same in teger v alue m ∈ Z > 0 app earing in ( 2.3 ), where t and ϵ are some p ositive num bers. The scaling n 2 m in σ n is motiv ated b y the following reasoning. Under our assumptions on V , the lo cal statistics of the classical Laguerre-type OP ensemble near the hard edge z = 0 ﬂuctuate on the scale O ( n − 2 ). Thus, in a lo cal co ordinate ζ ≈ n 2 x the statistics in the ζ -plane should behav e as O (1), that is, the v ariable ζ now detects the non trivial ﬂuctuations. Still with this same scaling, it follows from ( 2.6 ) that in the v ariable ζ w e ha v e Q ( x ) = O ( ζ m n − 2 m ) and therefore 1 1 + e − s − n 2 m Q ( x ) ≈ 1 1 + e − s − t ζ m . That is, with the scaling n 2 m in σ n and the v anishing condition ( 2.6 ) in place, the factor σ n in the deformed ensemble ( 2.1 ) pro duces a non trivial change of b ehavior in the ﬂuctuations at the local scale ζ , when compared to the classical Laguerre-t yp e OP ensemble. The positivity condition on Q is motiv ated by the follo wing reasoning. Poin t wise, the fact that Q > 0 implies that σ n ( x ) → 1 , n → ∞ , x > 0 . This means that σ n do es not aﬀect the global asymptotics of the random matrix model in in terv als of the form [ α, ∞ ) with α > 0, and only in a scale x = O ( n − 2 ) the statistics of the model should b ecome aﬀected. In principle, this global p ositivity condition on (0 , ∞ ) could b e replaced by p ositivit y enforced 6 L. MOLAG, G. SIL V A, AND L. ZHANG only near the hard edge x = 0, at the cost of more complicated analysis near p oin ts outside the hard edge. As w e men tioned earlier, since our goal is to sho w case no vel phenomena near the hard edge, w e opt for keeping this condition for the sake of simplicit y . Our main results explain these heuristic interpretations rigorously , computing all the limiting quan- tities in terms of integrable systems. T o state our results, w e introduce certain quan tities that allo w us to decode statistics of ( 2.1 ). The monic OPs P j ( x ) = P ( n ) j ( x | s ) = x j + · · · , j = 0 , 1 , 2 , . . . , for the w eight ω n ( · | s ) are uniquely determined b y Z ∞ 0 P j ( x ) x k ω n ( x | s )d x = 0 , k = 0 , . . . , j − 1 . The associated norming constants γ j ( s ) = γ ( n ) j ( s ) > 0, determined from 1 γ j ( s ) 2 = Z ∞ 0 P j ( x ) 2 ω n ( x | s )d x, are suc h that the sequence { γ j ( s ) P j , j = 0 , 1 , 2 , . . . } is orthonormal in L 2 ( ω n ( x | s )d x, [0 , ∞ )). With these quan tities in mind, we introduce the Christoﬀel-Darboux k ernel K n ( x, y ) = K n ( x, y | s ) . . = n − 1 X j =0 γ j ( s ) 2 P j ( x ) P j ( y ) , (2.7) and correlation kernel b K n ( x, y ) = b K n ( x, y | s ) . . = ω n ( x | s ) 1 / 2 K n ( x, y | s ) ω n ( y | s ) 1 / 2 . (2.8) All the quantities we just in troduced dep end smoothly on the parameter s , although sometimes we do not mak e explicit men tion to it in our notations. Also, as we will sho w, the limiting quan tities γ ( n ) j ( ∞ ) = lim s → + ∞ γ ( n ) j ( s ) and b K n ( · | ∞ ) = lim s → + ∞ b K n ( · | s ) mak e sense, and are the norming constants and correlation kernel for the undeformed w eigh t x α e − nV corresp onding to the classical Laguerre-type OP ensem ble. The relev ance of these quantities is the following. The norming constan ts relate to the partition function via the iden tit y (e.g., see [ 42 ]) Z n ( s ) = n ! n − 1 Y j =0 γ ( n ) j ( s ) − 2 . The ensem ble ( 2.1 ) itself forms a determinantal p oint pro cess: the join t distribution ( 2.1 ) may b e written in the determinan tal form det  b K n ( x i , x j | s )  n i,j =1 . F rom this determinan tal structure, statistics of the ensemb le ( 2.1 ) ma y be computed using the theory of determinan tal p oin t pro cesses [ 6 , 51 ]. In this regard, the multiplicativ e statistics L Q n ( s ) from ( 2.1 ) can b e computed in terms of a F redholm determinan t inv olving the kernel K n ( · | ∞ ) of the original Laguerre-t yp e ensem ble. Ho w ever, such an expression is not appropriate for asymptotics, as one w ould still need to analyse a full F redholm determinan t. Our starting p oin t will b e instead a trace-type CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 7 form ula that allows to compute L Q n ( s ) directly in terms of the diagonal of the deformed kernel K n ( · | s ). With L Q n , K n and ω n deﬁned in ( 2.5 ), ( 2.7 ) and ( 2.2 ), respectively , this form ula is log L Q n ( s ) = − Z ∞ s Z ∞ 0 K n ( x, x | u ) ∂ u ω n ( x | u )d x d u. (2.9) A v arian t of ( 2.9 ) app eared in [ 37 , Prop osition 9.1], where it was pro v en in the particular context of OPs o v er the full real line. Other v arian ts also app eared in [ 20 , End of Section 5.2] and [ 13 , Theorem 1.10] (see also [ 12 , 14 ]), whic h obtain their analogues exploring the Jacobi diﬀeren tiation iden tit y for F redholm determinan ts of so-called I IKS in tegrable op erators. The form ula as it app ears here is a particular case of [ 38 , Prop osition 4.1], whic h provides a deformation formula v alid for general OP ensem bles (including discrete ones), deformations of weigh ts beyond the case ( 2.2 ), and whic h relies only on the orthogonalit y of the polynomials P j used to construct the kernel K n ( · | s ). Thanks to ( 2.9 ), the ﬁrst step in understanding L Q n ( s ) as n → ∞ relies in obtaining asymptotics for K n ( · | s ). It turns out that in the large n limit, the ma jor contribution to the x -integration in ( 2.9 ) comes from a neighborho o d of the origin, precisely where K n has a nov el scaling limit. T o describ e this scaling limit, let us introduce the function σ Φ ( ζ ) = σ Φ ( ζ | s ) . . = 1 1 + e − s − ( − 1) m ζ m . (2.10) In this deﬁnition, the index Φ is introduced just for notational conv enience, to distinguish b etw een a generic function σ whic h will b e used later on. Theorem 2.3. Fix α > − 1 . F or ( ζ , s , x ) ∈ R × R × (0 , + ∞ ) , the nonlo c al diﬀer ential e quation ∂ 2 x Φ( ζ | s , x ) =  ζ + 4 α 2 − 1 4 x 2 + 2 e − π i α π Z ∞ s Z 0 −∞ Φ( ξ | u, x ) ∂ x Φ( ξ | u, x ) ∂ s σ Φ ( ξ | u )d ξ d u  Φ( ζ | s , x ) (2.11) admits a solution Φ = Φ( ζ | s , x ) satisfying for e ach x > 0 ﬁxe d, Φ( ζ ) =  1 + O ( ζ − 1 )  ζ − 1 / 4 √ 2 e x ζ 1 / 2 , ζ → + ∞ , Φ( ζ ) =  1 + O ( ζ − 1 )  √ 2 e π i α/ 2 | ζ | − 1 / 4 cos  x | ζ | 1 / 2 − π 4 − π α 2  , ζ → −∞ . F urthermor e, as x → 0 + , Φ( ζ | s , x ) = √ π x 1 / 2 I α ( x ζ 1 / 2 )  1 + O ( x 2+2 α (1 + x | ζ | 1 / 2 ))  , (2.12) uniformly for ζ ∈ R \ { 0 } and s ∈ R , wher e I α is the mo diﬁe d Bessel function of the ﬁrst kind with index α , and for ζ < 0 we must c onsider the + -b oundary value of I α . With the solution Φ just discussed, w e construct the kernel K α ( u, v ) . . = e − π i α p σ Φ ( − 4 u/ x 2 ) p σ Φ ( − 4 v / x 2 ) 2 π ( u − v ) ×  Φ  − 4 u x 2  ( ∂ x Φ)  − 4 v x 2  − Φ  − 4 v x 2  ( ∂ x Φ)  − 4 u x 2  , (2.13) v alid for u  = v , and extended b y contin uit y to u = v . Observe that Φ = Φ( · | s , x ), and therefore K α = K α ( · , · | s , x ). 8 L. MOLAG, G. SIL V A, AND L. ZHANG F or the next result, set c V . . = π 2 κ 2 0 , and identify x =  4 c V t 1 /m  1 / 2 , (2.14) where κ 0 > 0 is as in ( 2.4 ) and w e recall that t > 0 is as in ( 2.6 ). Theorem 2.4. Under Assumptions 2.1 and 2.2 and the identiﬁc ation ( 2.14 ) , the estimate 1 n 2 c V b K n  u c V n 2 , v c V n 2  = K α ( u, v | s , x ) + O (e − s n − κ + n − 2 ) , n → ∞ , holds uniformly for u, v in c omp act subsets of (0 , + ∞ ) , and uniformly for s ≥ s 0 with s 0 ∈ R ﬁxe d, and for any ﬁxe d κ ∈ (0 , 2) . The kernel b K n is the correlation kernel of the deformed matrix mo del ( 2.1 ), and in the limit s → + ∞ it degenerates to the regular (undeformed) matrix mo del with weigh t x α e − nV ( x ) . It w as shown in [ 56 ] that the latter con v erges to the Bessel kernel J α ( u, v ) := J α ( √ u ) √ v J ′ α ( √ v ) − √ uJ ′ α ( √ u ) J α ( √ v ) 2( u − v ) , u, v > 0 , (2.15) where J α is the Bessel function of the ﬁrst kind of order α . The uniformity of the bound ( 2.4 ) imme- diately yields the follo wing corollary . Corollary 2.5. The limit lim s → + ∞ K α ( u, v | s ) = J α ( u, v ) holds p ointwise for u, v > 0 , wher e J α is the Bessel kernel of or der α > − 1 deﬁne d in ( 2.15 ) . When scaled around the origin, the smallest eigenv alues of the mo del ( 2.1 ) con v erge to the Bessel p oin t pro cess (in the weak sense), which is the determinantal p oin t pro cess deﬁned by the kernel J α . Corollary 2.5 is the kernel conv ergence reﬂecting this property . Next, w e turn to the asymptotics for L Q n , for which we in tro duce L (Bes) α ( s , x ) . . = exp  − Z ∞ s Z ∞ 0 K α ( ζ , ζ | s = u )( ∂ s log σ Φ )  − 4 ζ x 2 | s = u  d ζ d u  . (2.16) The meaning of the upp er index (Bes) in the deﬁnition of L (Bes) α will be explained in a momen t. Theorem 2.6. Under Assumptions 2.1 and 2.2 and the identiﬁc ation ( 2.14 ) , the estimate log L Q n ( s ) = log L (Bes) α ( s , x ) + O  e − s n  , n → ∞ , holds uniformly for s ≥ s 0 with any ﬁxe d s 0 ∈ R . As men tioned earlier, the kernel b K n is the correlation kernel for the conditional thinning of the eigen v alues of ( 2.1 ). In the recent w ork [ 22 ], Claeys and the second-named author giv e conditions on the symbol σ n in ( 2.5 ) under whic h (1) the conditional thinning of a ﬁnite n random particle system con v erges to the conditional thinning of the large n limit of the particle system, and (2) multiplicativ e statistics of the ﬁnite n random particle system conv erge to the multiplicativ e statistics of their large n limit. CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 9 Mo dulo c hec king the tec hnical conditions required to apply the results from [ 22 ], regarding (1), Theorem 2.4 establishes that the kernel for the conditional thinning pro cess of the Bessel p oint pro cess asso ciated to the symbol σ Φ ( − 4 ζ x − 2 | s , x ) is precisely the k ernel K α . Regarding (2), Theorem 2.6 is sa ying that L (Bes) α ( s ) coincides with the multiplicativ e statistics of the Bessel p oin t pro cess for the same sym b ol, whic h also explains wh y we used the upp er index (Bes). This w ay , w e refer to K α as the conditional thinning kernel of the Bessel p oint pro cess, and L (Bes) α as a m ultiplicativ e statistic of the Bessel p oint pro cess. Checking the requirements for the application of the results in [ 22 ] is a tec hnical feat that we will not delv e into. With Φ as in Theorem 2.3 , in troduce p ( s , x ) . . = − 4 α 2 − 1 8 x + e − π i α 2 π Z ∞ s Z 0 −∞ Φ( ξ | s = u, x ) 2 ∂ s σ Φ ( ξ | s = u )d ξ d u. (2.17) W e ma y view ( 2.11 ) as a linear Sc hr¨ odinger equation with potential 2 ∂ x p , namely ∂ 2 x Φ( ζ | s , x ) = ( ζ + 2 ∂ x p ( s , x )) Φ( ζ | s , x ) . It turns out that we can relate the multiplicativ e statistic L (Bes) α directly to the potential p . Theorem 2.7. Fix α > − 1 . The estimate p ( s , x ) = − 4 α 2 − 1 8 x + 1 2 2 α +1 Γ( α + 1) 2 x 2 α +1 Z ∞ 0 u α e − s − u m 1 + e − s − u m d u + O ( x 2 α +3 ) , x → 0 + , (2.18) holds. In addition, the identity ∂ x  log L (Bes) α ( s , x )  = − p ( s , x ) − 4 α 2 − 1 8 x (2.19) is also true. Using ( 2.12 ), it is straightforw ard to sho w that log L (Bes) α go es to 0 as x → 0 + . In particular, Theorem 2.7 then pro vides the represen tation log L (Bes) α ( s , x ) = − Z x 0  p ( s , u ) + 4 α 2 − 1 8 u  d u. The celebrated β = 2 T racy-Widom distribution had its name coined thanks to the w ork of T racy and Widom [ 52 ], who show ed that the second log deriv ative of a m ultiplicativ e statistic of random matrices (a gap probability) at the soft edge could b e expressed in terms of the Hastings-McLeo d solution to the P ainlev ´ e I I equation. The analogue represen tation for gap probabilities of the Bessel kernel in terms of the Painlev ´ e V equation w as obtained shortly afterwards by T racy and Widom as w ell [ 53 ]. Sev eral diﬀeren t m ultiv ariate extensions hav e b een obtained in the past 20 y ears [ 1 , 3 , 7 , 8 , 54 , 57 ]. Theorem 2.7 is the analogue representation, showing that the second log deriv ative of L (Bes) α ma y b e expressed in terms of p which, in turn, is given in terms of the solution to the nonlo cal nonlinear equation ( 2.11 ). In the case m = 1, it is possible to obtain a PDE satisﬁed b y p directly . Theorem 2.8. Supp ose that m = 1 in ( 2.3 ) . Then the function p fr om ( 2.17 ) satisﬁes the PDE ∂ 4 sxxx p − 8 ∂ x p  ∂ 2 sx p + 1 2  − 4 ∂ 2 xx p  ∂ s p + x 2  = 0 . 10 L. MOLAG, G. SIL V A, AND L. ZHANG As said earlier, some of our results are in direct comparison with the recen t work [ 46 ] by Ruzza. Therein, and as mentioned earlier, Ruzza considers a class of multiplicativ e statistics for the Bessel p oin t process. His m ultiplicativ e statistics are coming from linear scalings of ﬁxed functions whic h are not necessarily con tin uous, and our family of statistics L (Bes) α reduces to a particular instance of his setup 1 only when m = 1. This w ay , Theorem 2.8 is a redisco v ery of [ 46 , Theorem 1.1–(ii)], the case m = 1 of ( 2.11 ) with b oundary condition ( 2.12 ) is equiv alen t to [ 46 , Theorem 1.1–(iii)], and the case m = 1 of ( 2.18 ) is the equiv alen t of [ 46 , Theorem 1.6]. Organization of the pap er. The remainder of this pap er is organized as follo ws. Section 3 is dev oted to the construction and analysis of a mo del RHP , which plays a cen tral role in the asymptotic analysis of OPs carried out later. This mo del problem gov erns the hard edge statistics. W e establish the solv abilit y of this mo del problem and derive the nonlo cal diﬀeren tial equations satisﬁed b y its solution. In Section 4 , we p erform a detailed asymptotic analysis of the mo del RHP with resp ect to a large parameter (whic h corresp onds to the system size n ), and analyze its limiting b eha vior in relev an t regimes. Section 5 wraps up the asymptotic consequences from Section 4 , in particular establishing certain asymptotic prop erties of the correlation kernel, and related integrals that will later app ear in the analysis of m ultiplicativ e statistics. W e also prov e main theorems on integrable equations in this section. Finally , Sections 6 and 7 contain the asymptotic analysis of the orthogonal p olynomials via the Deift-Zhou nonlinear steep est descent metho d, where the mo del problem and its prop erties established in the previous sections play a fundamen tal role. Ab out the notation. W e use D r ( z 0 ) to denote the disk on the complex plane centered at z 0 with radius r > 0, and D r = D r (0) for the particular case when z 0 = 0 is the origin. In general, we use b old capital letters Y , Ψ etc. to denote matrix-v alued functions. The letters ε, δ , η alwa ys denote p ositiv e constants that can be made arbitrarily small but are k ept ﬁxed, and w e alwa ys emphasize when they may dep end on external parameters. These small constan ts may hav e diﬀeren t v alues for diﬀeren t o ccurrences in the text. The set of strictly positive integers is Z > 0 , and we also use the notation Z ≥ 0 . . = Z > 0 ∪ { 0 } . When w e write that x → ∞ for some v ariable x , w e mean that x → + ∞ when x is real, or x → ∞ along an y direction of the complex plane in case x is allo w ed to assume v alues in C \ R as well. These t w o distinct cases will alw a ys b e clear from the con text and meaning of the v ariables in v olv ed. W e also use the following matrix notation. W e denote by I and 0 the iden tit y matrix and the null matrix, resp ectively , and by E ij the 2 × 2 matrix with 1 in the ( i, j )-en try and 0 in the remaining en tries. F or conv enience, w e set U 0 . . = 1 √ 2  1 i i 1  = 1 √ 2 ( I + i E 12 + i E 21 ) , σ 3 . . =  1 0 0 − 1  = E 11 − E 22 . (2.20) 1 With the identiﬁcation of v ariables ( x , s ) = ( x, − t ), our function p ( x , s ) is comparable with the function v ( x, t ) in [ 46 ] through the relation p ( x , s ) = − i  v ( x, t ) + xt 2  . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 11 In the course of the Riemann-Hilb ert analysis, we will use matrix norm notation. F or a matrix-v alued function M : U ⊂ C → C 2 × 2 , w e denote | M ( z ) | . . = max i,j =1 , 2 | M ij ( z ) | , where M ij stands for the ( i, j )-th entry of M . F or p ∈ [1 , ∞ ] and a curv e Γ ⊂ U , we also use the corresp onding L p (Γ) = L p (Γ , | d s | ) norm with resp ect to the arc length measure | d s | , ∥ M ∥ L p (Γ) . . = max i,j =1 , 2 ∥ M ij ∥ L p (Γ) . F or a matrix-v alued function M dep ending on a v ariable x ∈ R , it is conv enien t to in tro duce the notation ∆ x M ( x ) . . = M ( x ) − 1 M ′ ( x ) . (2.21) Under a change of v ariables x 7→ ζ = ζ ( x ), this operator transforms as ∆ x ( M ( ζ ( x )) = ζ ′ ( x ) M ( ζ ( x )) − 1 d M d ζ ( ζ ( x )) = ζ ′ ∆ ζ M ( ζ ) . (2.22) Ac kno wledgmen ts. G.S. thanks Mattia Cafasso, T om Claeys, Thomas Chouteau, Alfredo Dea ˜ no, and Giulio Ruzza, for v arious discussions related to this work. Parts of this pro ject w ere carried out during academic visits of G.S. to F udan Univ ersit y . He ac kno wledges the hospitality during these visits. This pro ject w as ﬁnalized while the three authors were taking part in the workshop Inte gr able Systems and R andom Matrix The ory , at Great Ba y Univ ersit y , Dongguan, China, in January 2026. They appreciate the hospitality and encouraging environmen t during the w orkshop. L.M. is supported by the UC3M gran t 2024/00002/007/001/023 “Local and global limits of complex- dimensional DPPs” and the gran t ID2024-155133NB-I00, “Orthogonality , Approximation, and In te- grabilit y: Applications in Classical and Quantum Sto chastic Pro cesses (OR TH-CQ)” by the Agencia Estatal de Inv estigaci´ on. G.S. ackno wledges supp ort by the S˜ ao Paulo Research F oundation (F APESP), Brazil, Pro cess Num- b er # 2019/16062-1, and b y the Brazilian National Council for Scientiﬁc and T ec hnological Developmen t (CNPq) under Grant # 306183/2023-4. L.Z. ackno wledges supp ort b y National Natural Science F oundation of China under Grant # 12271105 and “Shuguang Program” supp orted by Shanghai Education Developmen t F oundation and Shanghai Municipal Education Commission. 3. The model RHP As mentioned earlier, the main technical to ol we utilize is the RHP form ulation of OPs and associated quan tities. When w e p erform the asymptotic analysis of the OPs via the RHP metho d, at one of the core steps we need to construct an approximation to them near the origin, the so-called mo del RHP for the lo cal parametrix. F or usual Laguerre-t yp e OPs, such approximation is constructed using Bessel functions as originally developed in [ 40 ]; see [ 56 ] for details. Due to the nontrivial scaling of σ n near the origin, we need to use a diﬀeren t mo del problem, which we introduce and study in this section. This section is structured as follows. In Section 3.1 we in troduce the mo del problem itself. In Section 3.2 w e establish the solv abilit y of the mo del RHP , by means of v anishing-lemma-like arguments. In Section 3.3 , we connect the mo del problem with integrable systems; the conten t of the latter section ultimately leads to the diﬀerential equations from our main results. 12 L. MOLAG, G. SIL V A, AND L. ZHANG        J J J J J J J ^ -  0 Γ − Γ + Γ 0 S + S − S 0 r Figure 1. The con tours Γ ± , Γ 0 , and the sectors S ± , S 0 . 3.1. The mo del problem. Fix m ∈ Z > 0 , whic h will ha v e the same meaning as in ( 2.6 ), and set θ m . . = π  1 − 1 3 m  , Γ ± . . = (e ± i θ m ∞ , 0] , Γ 0 . . = ( −∞ , 0] , Γ . . = Γ + ∪ Γ 0 ∪ Γ − , (3.1) where the orientations of the arcs Γ ± , Γ 0 are tak en from ∞ to the origin. Also, introduce the sectors S 0 . . = { ζ ∈ C | − θ m < arg ζ < θ m } , S + . . = { ζ ∈ C | θ m < arg ζ < π } , S − . . = { ζ ∈ C | − π < arg ζ < − θ m } , (3.2) with principal v alue of the argument, that is, arg ζ ∈ ( − π , π ). W e refer to Figure 1 for an illustration of these arcs and sectors. Observ e that ∂ S 0 = Γ + ∪ Γ − , ∂ S + = Γ + ∪ Γ 0 , ∂ S − = Γ − ∪ Γ 0 . The v alue θ m is c hosen so that 1 2 | ζ | m ≤ ( − 1) m Re( ζ m ) ≤ | ζ | m for ζ ∈ S + ∪ S − , (3.3) with the upp er bound being attained only along ∂ S + ∩ ∂ S − = Γ 0 , and the low er b ound b eing attained only along ( ∂ S + ∪ ∂ S − ) \ Γ 0 = Γ + ∪ Γ − . Fix α > − 1 (which has the same meaning as in ( 2.2 )) and a function h : Γ → C . W e are interested in the following RHP . RHP 3.1. Find a 2 × 2 matrix-v alued function Ψ : C \ Γ → C 2 × 2 with the following prop erties. (i) Ψ is analytic on C \ Γ. (ii) The entries of Ψ ± are contin uous except p ossibly at the origin, and for ζ ∈ Γ \ { 0 } they satisfy the jump relation Ψ + ( ζ ) = Ψ − ( ζ ) J Ψ ( ζ ) , CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 13 with jump matrix J Ψ ( ζ ) . . =    I + (1 + e − h ( ζ ) ) e ± π i α E 21 , ζ ∈ Γ ± , 1 1 + e − h ( ζ ) E 12 − (1 + e − h ( ζ ) ) E 21 , ζ ∈ Γ 0 . (3.4) (iii) F or some matrix-v alued function Ψ 1 whic h is indep endent of ζ , the following expansion is v alid, Ψ ( ζ ) =  I + Ψ 1 ζ + O ( ζ − 2 )  ζ − σ 3 / 4 U 0 e 2 ζ 1 / 2 σ 3 , ζ → ∞ , (3.5) with the principal branc hes of the fractional p ow ers, and where U 0 and σ 3 are deﬁned in ( 2.20 ). (iv) The en tries of Ψ ± b elong to L 2 loc (Γ , | d s | ). Condition (ii) asks for con tin uit y of the b oundary v alues Ψ ± except p ossibly at the origin, so Con- dition (iv) is an additional requiremen t only ab out the b ehavior of Ψ near the origin. In the statemen t of RHP 3.1 ab ov e, w e ha v e used the function h , but in fact the jump ma y be written simply in terms of the function σ giv en b y σ : Γ → C , σ ( ζ ) . . = 1 1 + e − h ( ζ ) , (3.6) namely as J Ψ ( ζ ) =        I + e ± π i α σ ( ζ ) E 21 , ζ ∈ Γ ± , σ ( ζ ) E 12 − 1 σ ( ζ ) E 21 , ζ ∈ Γ 0 . Historically , factors of the form (1 + e − h ) − 1 ha v e app eared in the so-called ﬁnite temp erature de- formations of random matrix mo dels, and partly for that reason oftentimes it is conv enien t to state conditions/results in terms of h . Ho w ev er, in what follows w e ma y use either σ or h , dep ending on the con v enience of the situation. Ob viously , Ψ dep ends on h , α , and when we need to make this dep endence explicit w e write Ψ = Ψ ( · | h ) , Ψ ( · | α ) , Ψ ( · | h , α ), etc. F or the RHP ab ov e to b e well-posed, we should imp ose conditions on h . Global conditions on h will b e imp osed in a moment, but it is conv enient to write the condition (iv) in a more explicit manner when h is analytic at the origin, as giv en by the next result. T o pro ceed, w e in tro duce a ( ζ ) = a α ( ζ ) . . =        1 2i sin( απ ) , α / ∈ Z , α > − 1 , e π i α 2 π i log ζ , α ∈ Z ≥ 0 , (3.7) for ζ ∈ C \ ( −∞ , 0]. It is easily seen that a satisﬁes the jump condition a + ( ζ ) e π i α − a − ( ζ ) e − π i α = 1 , ζ < 0 , (3.8) for an y α > − 1. This function will b e used several times throughout the text. Prop osition 3.2. Supp ose that h is analytic in a neighb orho o d of the origin and 1 + e − h never vanishes ne ar the origin. Then Condition (iv) in RHP 3.1 implies that for some ε > 0 , the identities Ψ ( ζ ) = Ψ 0 ( ζ ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 ) σ ( ζ ) − σ 3 / 2 ( I − σ ( ζ ) − 1 ( χ S + ( ζ ) e π i α − χ S − ( ζ ) e − π i α ) E 21 ) = Ψ 0 ( ζ ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 )( I − ( χ S + ( ζ ) e π i α − χ S − ( ζ ) e − π i α ) E 21 ) σ ( ζ ) − σ 3 / 2 (3.9) 14 L. MOLAG, G. SIL V A, AND L. ZHANG hold for ζ ∈ D ε , wher e Ψ 0 ( ζ ) is analytic on D ε , al l the r o ots ar e princip al br anches, and χ S ± is the char acteristic function of the set S ± . Pr o of. The second equality in ( 3.9 ) follows from a simple algebraic manipulation, whic h we now use without further justiﬁcation. W e use the ﬁrst iden tit y in ( 3.9 ) as a deﬁning relation for Ψ 0 and pro v e that it is an analytic function near the origin. Note that the factors ζ α/ 2 , σ − 1 / 2 and a are analytic and non-v anishing on a set of the form D ε \ ( − ε, 0], from which it follo ws that Ψ 0 is piecewise analytic, with jumps along Γ. A direct calculation, also using the analyticity of ζ α/ 2 , σ − 1 / 2 , a aw ay from the negativ e axis, shows that Ψ 0 is analytic across Γ + and Γ − as w ell. F or ζ ∈ Γ 0 \ { 0 } , w e compute Ψ 0 , − ( ζ ) − 1 Ψ 0 , + ( ζ ) = I + | ζ | α  1 − a + ( ξ ) e π i α + a − ( ζ ) e − π i α  E 12 . In virtue of ( 3.8 ), the righ t-hand side ab ov e is equal to I , and w e conclude that Ψ 0 is analytic across Γ 0 as w ell. The arguments ab ov e show that ζ = 0 is an isolated singularity of Ψ 0 . Because we are assuming that the singularities of the en tries of Ψ at the origin are square-in tegrable and α > − 1, we see that the en tries of Ψ 0 , ± b elong to L 1 , implying that Ψ 0 has a remov able singularit y at ζ = 0. □ R emark 3.3 . The previous prop osition requires that σ − 1 = 1 + e − h is analytic and non-v anishing near the origin, but not necessarily h itself. It is thus immediate to see that its conclusion also holds for the c hoice h = + ∞ , which leads to 1 + e − h ≡ 1; for this choice the RHP for Ψ reduces to a RHP for Bessel functions (see App endix A b elow), which is kno wn and will be used later. ▷ F or the remainder of this section we are interested in tw o indep enden t asp ects. First of all, under some additional conditions on h w e will pro v e that the model problem is solv able. Second, for certain particular choices of h , we explain ho w our mo del problem is connected to a family of in tegrable systems. 3.2. Solv abilit y of the mo del problem. Our goal in this section is to pro ve that under some conditions on h , the RHP 3.1 for Ψ alw ays admits a solution. Recall the contours Γ ± , Γ 0 and the sectors S 0 , S ± deﬁned in ( 3.1 ) and ( 3.2 ), resp ectively , w e assume the follo wing. Assumption 3.4. The function h in ( 3.4 ) satisﬁes the follo wing conditions. (i) h : G → C is well-deﬁned and analytic in some op en set G that contains the closure of the union of sectors S + ∪ S − . (ii) h is real-v alued along ( −∞ , 0). (iii) The asso ciated function σ from ( 3.6 ) extends to G with a b ounded con trol. More precisely , σ : G → C , σ ( ζ ) . . = 1 1 + e − h ( ζ ) , remains bounded as ζ → ∞ along G , and furthermore σ ( ζ ) = 1 + O ( ζ − 2 ) , as ζ → ∞ in a neigh b orho o d of ( −∞ , 0). The ﬁnal goal of this section is to pro v e the following result. Theorem 3.5. Under Assumption 3.4 , RHP 3.1 for Ψ admits a unique solution. CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 15 The uniqueness claim in Theorem 3.5 follows in a simple standard wa y in RHP theory [ 27 ]. W e will establish the existence of the solution Ψ b y means of the tec hnique of a v anishing lemma. F ollowing the general scheme laid out in [ 30 , 35 , 59 ], a k ey step in the argumen t is to sho w that the “homogeneous” v ersion of the RHP has only the trivial solution. Over here, the relev an t homogeneous version of the RHP 3.1 is the following. RHP 3.6. Find a 2 × 2 matrix-v alued function Ψ H : C \ Γ → C 2 × 2 with the following prop erties. (i) Ψ H is analytic on C \ Γ. (ii) The entries of Ψ H ± are contin uous except p ossibly at ζ = 0, and for ζ ∈ Γ \ { 0 } they satisfy the jump relation Ψ H + ( ζ ) = Ψ H − ( ζ ) J Ψ ( ζ ) , with the same jump matrix J Ψ as in ( 3.4 ). (iii) As ζ → ∞ , Ψ H ( ζ ) = O ( ζ − 1 ) ζ − σ 3 / 4 U 0 e 2 ζ 1 / 2 σ 3 . (iv) As ζ → 0, it behav es as Ψ H ( ζ ) = Ψ H 0 ( ζ ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 )( I − ( χ S + ( ζ ) e π i α − χ S − ( ζ ) e − π i α ) E 21 ) σ ( ζ ) − σ 3 / 2 , (3.10) where Ψ H 0 is analytic in a neigh b orho o d of the origin, a and σ are deﬁned in ( 3.7 ) and ( 3.6 ), resp ectiv ely . As men tioned, from standard RHP theory it follo ws that Theorem 3.5 is prov ed once we establish the next lemma, whic h is usually referred to as a v anishing lemma. Lemma 3.7. Under Assumption 3.4 , the unique solution to RHP 3.6 is the trivial solution Ψ H ≡ 0 . This wa y , for the remainder of this section our work will b e fo cused tow ards proving Lemma 3.7 , whic h will b e established follo wing a standard route of making certain transformations of the homogeneous RHP , and then applying the theory of analytic functions to pro v e that the entries of a transformed v ersion of Ψ H are all identically zero. As a ﬁrst step, w e explore that the function h is analytic on a neigh b orho o d of S + ∪ S − , and transform Φ H ( ζ ) . . = Ψ H ( ζ )  I + σ ( ζ ) − 1  e π i α χ S + ( ζ ) − e − π i α χ S − ( ζ )  E 21  e − 2 ζ 1 / 2 σ 3 . This transformation has the eﬀect of removing the jumps outside the real axis, and also to remov e the exp onen tial b eha vior from the asymptotics at ∞ . More precisely , a direct calculation shows that the function Φ H solv es the following RHP . RHP 3.8. Find a 2 × 2 matrix-v alued function Φ H : C \ ( −∞ , 0] → C 2 × 2 with the following prop erties. (i) Φ H is analytic on C \ ( −∞ , 0]. (ii) The entries of Φ H ± are con tinuous except p ossibly at the origin, and for ζ ∈ ( −∞ , 0) they satisfy the jump relation Φ H + ( ζ ) = Φ H − ( ζ )  e ( π i α − 4i | ζ | 1 / 2 ) σ 3 + σ ( ζ ) E 12  . (3.11) (iii) As ζ → ∞ , Φ H ( ζ ) = O ( ζ − 1 ) ζ − σ 3 / 4 U 0 O (1) = O ( ζ − 3 / 4 ) . 16 L. MOLAG, G. SIL V A, AND L. ZHANG (iv) As ζ → 0, it behav es as Φ H ( ζ ) = Ψ H 0 ( ζ ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 ) O (1) , where Ψ H 0 is analytic in a neighborho o d of the origin as given in ( 3.10 ). Next, set X H ( ζ ) . . = Φ H ( ζ )      0 − 1 1 0 ! , Im ζ > 0 , I , Im ζ < 0 . With this transformation, we mov e the diagonal entries in ( 3.11 ) to oﬀ-diagonal en tries, and the matrix X H is a solution to the follo wing RHP . RHP 3.9. Find a 2 × 2 matrix-v alued function X H : C \ R → C 2 × 2 with the following prop erties. (i) X H is analytic on C \ R . (ii) The entries of X H ± are con tinuous except p ossibly at the origin, and for ζ ∈ R they satisfy the jump relation X H + ( ζ ) = X H − ( ζ ) J X H ( ζ ) , (3.12) with jump matrix J X H ( ζ ) =            0 − 1 1 0 ! , ζ > 0 , σ ( ζ ) − e π i α − 4i | ζ | 1 / 2 e − π i α +4i | ζ | 1 / 2 0 ! , ζ < 0 . (3.13) (iii) As ζ → ∞ , X H ( ζ ) = O ( ζ − 3 / 4 ) . (iv) As ζ → 0, it behav es as X H ( ζ ) = Ψ H 0 ( ζ ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 ) O (1) , where Ψ H 0 is analytic in a neighborho o d of the origin as given in ( 3.10 ). W e are now ready to pro v e Lemma 3.7 and thus conclude Theorem 3.5 . Pr o of of L emma 3.7 . Denote by M ∗ the adjoint (complex transp ose) of a matrix M . The jump matrix J X H enjo ys the symmetry prop erties J X H ( ζ ) + J X H ( ζ ) ∗ = ( 0 , ζ > 0 , 2 σ ( ζ ) E 11 , ζ < 0 . (3.14) T o explore these symmetries, let us set H ( ζ ) . . = X H ( ζ ) X H ( ζ ) ∗ , ζ ∈ C \ R . Then H is analytic on C \ R and has the behavior H ( ζ ) = O ( ζ − 3 / 2 ) , ζ → ∞ . Using ( 3.7 ) and the deﬁnitions of H and X H , it also follo ws that H ( ζ ) = O ( ζ α (1 + c α log ζ )) , ζ → 0 , CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 17 for some constant c α dep ending on α with c α = 0 if α / ∈ Z . In particular, this transformation regularizes the b eha vior at the origin and at ∞ , to an integrable b ehavior. Using Cauch y’s Theorem, w e then obtain 0 = Z R H + ( ζ ) d ζ = Z R X H − ( ζ ) J X H ( ζ )( X H ( ζ ) ∗ ) + d ζ = Z R X H − ( ζ ) J X H ( ζ ) X H − ( ζ ) ∗ d ζ . Adding the in tegral on the right-hand side with its adjoint and then using ( 3.14 ) we obtain the identit y Z 0 −∞ X H − ( ζ ) E 11 X H − ( ζ ) ∗ σ ( ζ ) d ζ = 0 . W riting en try-wise, and using that σ > 0 along the real axis, we hav e X H 11 , − ( ζ ) = X H 21 , − ( ζ ) = 0 , ζ < 0 . With the aid of the jump relation ( 3.12 ) and ( 3.13 ) satisﬁed b y X H , w e further conclude that X H 12 , + ( ζ ) = X H 22 , + ( ζ ) = 0 , ζ < 0 . Moreo v er, with the principal branch of ζ 1 / 2 , the jump relation of X H also implies that for j = 1 , 2, the function ζ 7→ ( X H j 2 ( ζ ) , ζ ∈ S + , − e π i α +4 ζ 1 / 2 X H j 1 ( ζ ) , ζ ∈ S − , is analytic on a neighborho o d of the interv al ( −∞ , 0), and by the identities w e just obtained it v anishes on this interv al. Hence, these functions are identically zero on S ± , and b y analytic con tinuation we conclude that X H 11 ( ζ ) = X H 21 ( ζ ) = 0 , Im ζ < 0 , and X H 12 ( ζ ) = X H 22 ( ζ ) = 0 , Im ζ > 0 . (3.15) As a ﬁnal step, we now improv e these iden tities to the v anishing of X H in the whole complex plane. F or that, we account for ( 3.15 ) and write the jump condition ( 3.12 ) and ( 3.13 ) for X H as X H j 1 , + ( ζ ) =  X H j 2 ( ζ ) e − π i α − 4 ζ 1 / 2  − , ζ < 0 , X H j 1 , + ( ζ ) = X H j 2 , − ( ζ ) , ζ > 0 , where in the ﬁrst line w e use the principal branc h of the square-root and j = 1 , 2. Such jump conditions motiv ate us to deﬁne the scalar functions g H j ( ζ ) . . = ( X H j 1 ( ζ ) , Im ζ > 0 , X H j 2 ( ζ ) , Im ζ < 0 . (3.16) The prop erties of X H w e just discussed then imply that g H j is a solution to the following scalar RHP . RHP 3.10. F or j = 1 , 2, the scalar function g H j deﬁned in ( 3.16 ) satisﬁes the follo wing conditions. (i) g H j is analytic on C \ ( −∞ , 0]. (ii) F or ζ < 0, it satisﬁes the jump relation g H j, + ( ζ ) = g H j, − ( ζ ) e − π i α +4i | ζ | 1 / 2 . (3.17) (iii) As ζ → ∞ , g H j ( ζ ) = O ( ζ − 3 / 4 ) . 18 L. MOLAG, G. SIL V A, AND L. ZHANG (iv) As ζ → 0, g H 1 ( ζ ) = O ( ζ α/ 2 (1 + c α log ζ )) , g H 2 ( ζ ) = O ( ζ − α/ 2 (1 + c α log ζ )) , where c α  = 0 only for α ∈ Z ≥ 0 . Let us now redeﬁne b g H j ( ζ ) = g H j ( ζ 2 ) , Re ζ > 0 . Then b g H j is analytic on Re ζ > 0 and it admits an analytic contin uation to a sector containing the imaginary axis. In fact, the jump condition ( 3.17 ) for g H j along ( −∞ , 0) implies that for any θ ∈ (0 , π / 2), the function b g H j ( ζ ) =      g H j ( ζ 2 ) , Re ζ > 0 , g H j ( ζ 2 ) e − π i α +4 ζ , π 2 < arg ζ < π 2 + θ , g H j ( ζ 2 ) e π i α +4 ζ , − π 2 − θ < arg ζ < − π 2 , is analytic across i R \ { 0 } , and satisﬁes b g H j ( ζ ) = O ( ζ − 3 / 2 ) , ζ → ∞ with − π 2 − θ < arg ζ < π 2 + θ , as w ell as b g H 1 ( ζ ) = O ( ζ α (1 + c α log ζ )) , b g H 2 ( ζ ) = O ( ζ − α (1 + c α log ζ )) , ζ → 0 . Finally , let us further in troduce e g H j ( ζ ) =  ζ 1 + ζ  | α | +1 b g H j ( ζ ) . It follows that e g H j is analytic on − π 2 − θ < arg ζ < π 2 + θ , it go es to 0 as ζ → ∞ along the same sector, and it is also bounded as ζ → 0. Applying Phragm´ en-Lindel¨ of principle [ 49 , Section 5.1, in particular Exercise 5.1.2], it follo ws that e g H j ≡ 0. T racing back the transformations, this shows that the entries of X H are all identically zero, as w e w an ted. □ 3.3. Diﬀeren tial equations for the mo del problem with some particular data. Consider the mo del problem with the particular choice h ( ζ ) = h ∞ ( ζ | s , u ) . . = s + u ( − 1) m ζ m , s ∈ R , u > 0 . (3.18) The index ∞ is used b ecause later this c hoice h ∞ will appear as a limiting data. Let Ψ ∞ ( ζ ) . . = Ψ ( ζ | h = h ∞ , α ) (3.19) b e the asso ciated solution to the RHP 3.1 . Thanks to Theorem 3.5 , the solution Ψ ∞ exists uniquely . The goal of this subsection is to connect certain quan tities asso ciated to Ψ ∞ to integrable systems. Recall that σ Φ w as in troduced in ( 2.10 ). It may b e alternativ ely expressed as σ Φ ( ζ ) = 1 1 + e − h ∞ ( ζ / u 1 /m ) . (3.20) F or the following calculations, it turns out to b e conv enien t to collapse the jumps outside the real axis do wn to the negative axis. This pro cess is p erformed through the mo diﬁcation of Ψ ∞ giv en b y Φ ∞ ( ζ ) . . = u − σ 3 / (4 m ) Ψ ∞ ( ζ / u 1 /m )  I + σ Φ ( ζ ) − 1  e π i α χ S + ( ζ ) − e − π i α χ S − ( ζ )  E 21  , ζ ∈ C \ Γ , (3.21) where w e recall that the sectors S ± are displa y ed in Figure 1 . It then follows from RHP 3.1 for Ψ ∞ and ( 3.19 ) that Φ ∞ satisﬁes the following RHP . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 19 RHP 3.11. Find a 2 × 2 matrix-v alued function Φ ∞ : C \ ( −∞ , 0] → C 2 × 2 with the follo wing prop erties. (i) Φ ∞ is analytic on C \ ( −∞ , 0]. (ii) The en tries of Φ ∞ , ± are con tin uous along ( −∞ , 0) and they are related by Φ ∞ , + ( ζ ) = Φ ∞ , − ( ζ ) J Φ ∞ ( ζ ) , with J Φ ∞ ( ζ ) . . = e π i α σ 3 + σ Φ ( ζ ) E 12 . (3.22) (iii) There exists a matrix Φ ∞ , 1 , independent of ζ , suc h that as ζ → ∞ , Φ ∞ ( ζ ) =  I + Φ ∞ , 1 ζ + O ( ζ − 2 )  ζ − σ 3 / 4 U 0 e x ζ 1 / 2 σ 3  I + (e π i α χ S + ( ζ ) − e − π i α χ S − ( ζ )) E 21  , (3.23) where w e hav e set x . . = 2 u − 1 2 m > 0 . (3.24) (iv) There exists a matrix-v alued function Φ ∞ , 0 whic h is analytic near ζ = 0 for whic h Φ ∞ ( ζ ) = Φ ∞ , 0 ( ζ ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 ) σ Φ ( ζ ) − σ 3 / 2 , ζ → 0 , (3.25) where a is as in ( 3.7 ). F or later use, w e record that the co eﬃcient Φ ∞ , 1 is related with the co eﬃcien t Ψ 1 = Ψ ∞ , 1 ( h = h ∞ ) app earing in ( 3.5 ) through the iden tit y Φ ∞ , 1 = u 1 /m u − σ 3 / (4 m ) Ψ ∞ , 1 u σ 3 / (4 m ) = 4 x 2 2 − σ 3 / 2 x σ 3 / 2 Ψ ∞ , 1 x − σ 3 / 2 2 σ 3 / 2 . There are tw o ma jor reasons for the c hange Ψ ∞ 7→ Φ ∞ . First, with the introduction of the charac- teristic functions in the right-most side of ( 3.21 ) the matrix Φ ∞ has jumps only along ( −∞ , 0). This structure will b e useful later when deriving the men tioned nonlo cal equation. Second, as a consequence of the change of v ariables ζ 7→ ζ / u 1 /m in ( 3.21 ), we obtain that the jump matrix J Φ ∞ for Φ ∞ is indep enden t of the external parameter x . As a consequence of the latter prop erty , we obtain a diﬀeren tial equation for Φ ∞ in the v ariable x . The argument is standard and go es as follo ws. Both Φ ∞ and its deriv ativ e ∂ x Φ ∞ ha v e the same jump relations, and therefore the matrix A ( ζ ) . . = ∂ x Φ ∞ ( ζ ) Φ ∞ ( ζ ) − 1 is analytic across ( −∞ , 0). In other w ords, A has an isolated singularity at ζ = 0, and it is analytic elsewhere in C . F rom the explicit b eha vior of Φ ∞ at ζ = 0 in ( 3.25 ) we see that the singularit y of A at ζ = 0 is remov able, that is, all the en tries of A are entire functions. F rom the b ehavior of Φ ∞ at ζ = ∞ in ( 3.23 ) w e obtain an explicit expression for A in terms of Φ ∞ , 1 . W riting Φ ∞ , 1 =  q − i p i r − q  , with q = q ( s , x ) , r = r ( s , x ) and p = p ( s , x ) = 2i x ( Ψ ∞ , 1 ) 12 , (3.26) w e obtain ∂ x Φ ∞ ( ζ ) = A ( ζ ) Φ ∞ ( ζ ) , with A ( ζ ) . . =  p − i i( ζ − 2 q ) − p  . (3.27) Moreo v er, the 1 /ζ term in the large ζ -expansion of ∂ x Φ ∞ ( ζ ) Φ ∞ ( ζ ) − 1 m ust v anish, and the (1 , 2)-en try therein giv es us p 2 = 2 q + ∂ x p . (3.28) Although the matrix A is rather simple-looking, it is conv enien t to transform Φ ∞ a little further, in order to obtain an ODE with a matrix coeﬃcient with v anishing diagonal. T o that end, we transform Υ ( ζ ) . . = ( I + i p E 21 ) Φ ∞ ( ζ ) . (3.29) A direct calculation using ( 3.27 ) and ( 3.28 ) then sho ws that Υ satisﬁes the ODE ∂ x Υ ( ζ ) = B ( ζ ) Υ ( ζ ) , B ( ζ ) . . = B 1 ζ + B 0 , B 1 . . = i E 21 , B 0 . . = 2i ∂ x p E 21 − i E 12 . (3.30) 20 L. MOLAG, G. SIL V A, AND L. ZHANG This new ODE implies in particular the relation Υ 2 k ( ζ ) = i ∂ x Υ 1 k ( ζ ) , k = 1 , 2 . (3.31) F urthermore, taking one more ∂ x -deriv ative in ( 3.30 ), lo oking en trywise, and setting Υ 1 k ( ζ ) = Φ k ( ζ ) , Φ . . = Φ 1 = ( Φ ∞ ) 11 , (3.32) w e obtain ∂ 2 x Φ k ( ζ ) = ( ζ + 2 ∂ x p ) Φ k ( ζ ) , k = 1 , 2 . (3.33) The function Φ pla ys a ma jor role in our results. F or all the calculations that follo ws, we understand Φ( ζ ) = Φ + ( ζ ) for ζ < 0 , ha ving in mind also that b y the jump condition ( 3.22 ), Φ( ζ ) = Φ + ( ζ ) = e π i α Φ − ( ζ ) , ζ < 0 . (3.34) Observ e that the v ery deﬁnition of Φ in ( 3.32 ) in terms of Φ ∞ , when com bined with ( 3.23 ), yields the asymptotic behaviors Φ( ζ ) =  1 + O ( ζ − 1 )  ζ − 1 / 4 √ 2 e x ζ 1 / 2 , ζ → + ∞ , (3.35) and Φ( ζ ) =  1 + O ( ζ − 1 )  √ 2 e π i α/ 2+ π i / 4 ζ − 1 / 4 + cos  x | ζ | 1 / 2 − π 4 − π α 2  , ζ → −∞ . (3.36) In summary , equation ( 3.30 ) is a ﬁrst order ODE for Υ , which yields a second order ODE for its en tries, given by ( 3.33 ) and the relation ( 3.31 ). F ollowing the usual dressing metho d for RHPs, one w ould b e tempted in obtaining a similar ODE for Υ in the ζ v ariable, and from which a Lax pair whose compatibility condition should yield scalar ODEs/PDEs of interest. Ho w ev er, the k ey feature for obtaining ( 3.30 ) was the fact that the jump matrix J Υ = J Φ ∞ for Υ is indep endent of x , and b ecause it is not constant as a function of ζ or s it is not p ossible to obtain the ODE in the ζ or s v ariables. T o o v ercome this issue, w e follo w a mo diﬁcation of the dressing metho d 2 . Start b y deﬁning W 0 = i ∂ s p E 21 , whic h is introduced so that ∂ s Υ ( ζ ) Υ ( ζ ) − 1 − W 0 = O ( ζ − 1 ) , ζ → ∞ . F or later reference, w e state the expansion ∂ s Υ ( ζ ) Υ ( ζ ) − 1 = W 0 + 1 ζ W − 1 + O ( ζ − 2 ) , ζ → ∞ , (3.37) with W − 1 . . =  ∂ s q − p ∂ s p − i ∂ s p i  p 2 ∂ s p − 2 p ∂ s q − ∂ s r  − ∂ s q + p ∂ s p  , as w ell as ∂ s Υ ( ζ ) Υ ( ζ ) − 1 − W 0 = a ( ζ ) ζ α ∂ s log σ Φ ( ζ )( I + i p E 21 ) Φ ∞ , 0 (0) E 12 Φ ∞ , 0 (0) − 1 ( I − i p E 21 ) + O (1) , ζ → 0 . (3.38) Mo ving further, introduce V ( ζ ) . . = Υ ( ζ ) − 1 ∂ s Υ ( ζ ) − Υ ( ζ ) − 1 W 0 Υ ( ζ ) , ζ ∈ C \ ( −∞ , 0] , 2 W e thank Thomas Chouteau for introducing this technique to us. CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 21 so that ∂ s Υ ( ζ ) = W 0 Υ ( ζ ) + Υ ( ζ ) V ( ζ ) , ζ ∈ C \ ( −∞ , 0] . (3.39) Then, the pro duct ΥVΥ − 1 is analytic on C \ ( −∞ , 0], and thanks to ( 3.37 )–( 3.38 ), Υ ( ζ ) V ( ζ ) Υ ( ζ ) − 1 = ∂ s Υ ( ζ ) Υ ( ζ ) − 1 − W 0 = ( O ( ζ − 1 ) , ζ → ∞ , O (1 + a ( ζ ) ζ α ) , ζ → 0 . (3.40) W e no w compute additiv e jump relations for ΥVΥ − 1 along ( −∞ , 0). The matrix W 0 is constant and therefore it has no jumps, so that ( Υ ( ζ ) V ( ζ ) Υ ( ζ ) − 1 ) + − ( Υ ( ζ ) V ( ζ ) Υ ( ζ ) − 1 ) − = ( ∂ s Υ ( ζ ) Υ ( ζ ) − 1 ) + − ( ∂ s Υ ( ζ ) Υ ( ζ ) − 1 ) − = Υ + ( ζ ) J Υ ( ζ ) − 1 ∂ s J Υ ( ζ ) Υ + ( ζ ) − 1 = e − π i α ∂ s σ Φ ( ζ ) Υ + ( ζ ) E 12 Υ + ( ζ ) − 1 . F rom ( 3.31 ), ( 3.32 ) and the identit y det Υ ≡ 1 w e compute ΥE 12 Υ − 1 , ﬁnally obtaining ( Υ ( ζ ) V ( ζ ) Υ ( ζ ) − 1 ) + − ( Υ ( ζ ) V ( ζ ) Υ ( ζ ) − 1 ) − = e − π i α ∂ s σ Φ ( ζ )  − iΦ( ζ ) ∂ x Φ( ζ ) Φ( ζ ) 2 ( ∂ x Φ( ζ )) 2 iΦ( ζ ) ∂ x Φ( ζ )  , (3.41) where in the abov e and in what follows we recall that we understand Φ( ζ ) = Φ + ( ζ ) for ζ < 0. The jump relation ( 3.41 ) yields a scalar additive RHP for ΥVΥ − 1 along the jump contour ( −∞ , 0). When coupled with the b oundary conditions ( 3.40 ) on the endp oin ts ∞ , 0, it can b e uniquely solved using Plemelj’s formula. In order to v erify suc h solution, w e still need to understand the b eha vior of its jump, given by the righ t-hand side of ( 3.41 ), near the endp oints ∞ , 0, which we do next. With Φ ∞ , 0 b eing the analytic function on the righ t-hand side of ( 3.25 ), denote Φ 0 ( ζ ) . . = ( Φ ∞ , 0 ) 11 . A direct calculation using ( 3.25 ) and ( 3.29 ) shows that Φ( ζ ) ∼ Φ 0 ( ζ ) ζ α/ 2 σ Φ ( ζ ) − 1 / 2 , ζ → 0 . Therefore, ∂ s σ Φ ( ζ )  − iΦ( ζ ) ∂ x Φ( ζ ) Φ( ζ ) 2 ( ∂ x Φ( ζ )) 2 iΦ( ζ ) ∂ x Φ( ζ )  ∼ ζ α ∂ s σ Φ ( ζ ) σ Φ ( ζ )  − iΦ 0 ( ζ ) ∂ x Φ 0 ( ζ ) Φ 0 ( ζ ) 2 ( ∂ x Φ 0 ( ζ )) 2 iΦ 0 ( ζ ) ∂ x Φ 0 ( ζ )  , and together with the fact that ζ 7→ ∂ s log σ Φ ( ζ ) is bounded along the real axis (see ( 2.10 )), we learn ∂ s σ Φ ( ζ )  − iΦ( ζ ) ∂ x Φ( ζ ) Φ( ζ ) 2 ( ∂ x Φ( ζ )) 2 iΦ( ζ ) ∂ x Φ( ζ )  = O (1 + ζ α ) , ζ → 0 . Lik ewise, from the behavior ( 3.23 ) w e get the rough estimate Φ( ζ ) = O ( ζ − 1 / 4 ) as ζ → −∞ , and from the explicit expression for σ Φ in ( 2.10 ) we learn that ∂ s σ Φ ( ζ )  − iΦ( ζ ) ∂ x Φ( ζ ) Φ( ζ ) 2 ( ∂ x Φ( ζ )) 2 iΦ( ζ ) ∂ x Φ( ζ )  = O (e −| ζ | m ) , ζ → −∞ . This estimate is not necessarily uniform in s but it is suﬃcien t for our purp oses. These last tw o estimates show that the righ t-hand side of ( 3.41 ) is in tegrable o ver the negativ e axis, so its Cauc h y transform is w ell-deﬁned. They also ensure that this Cauc h y transform satisﬁes b oth lo cal behaviors ( 3.40 ). As a consequence, we ﬁnally learn that Υ ( ζ ) V ( ζ ) Υ − 1 ( ζ ) = e − π i α 2 π i Z 0 −∞  − iΦ( ξ ) ∂ x Φ( ξ ) Φ( ξ ) 2 ( ∂ x Φ( ξ )) 2 iΦ( ξ ) ∂ x Φ( ξ )  ∂ s σ Φ ( ξ )d ξ ξ − ζ , ζ ∈ C \ ( −∞ , 0] . 22 L. MOLAG, G. SIL V A, AND L. ZHANG Let us combine what w e hav e so far. Equations ( 3.30 ) and ( 3.39 ) form a Lax pair, ∂ x Υ ( ζ ) = B ( ζ ) Υ ( ζ ) and ∂ s Υ ( ζ ) = Υ ( ζ ) V ( ζ ) + W 0 Υ ( ζ ) . The compatibilit y b etw een these t w o equations reads ∂ s B ( ζ ) − ∂ x W 0 + [ B ( ζ ) , W 0 ] = [ Υ ( ζ ) V ( ζ ) Υ ( ζ ) − 1 , B ( ζ )] + ∂ x ( Υ ( ζ ) V ( ζ ) Υ ( ζ ) − 1 ) . (3.42) The terms B and W 0 are matrix-v alued rational functions in ζ , whereas ΥVΥ − 1 - alb eit more com- plicated - admits an asymptotic expansion as ζ → ∞ of the form Υ ( ζ ) V ( ζ ) Υ − 1 ( ζ ) ∼ − ∞ X k =1 1 ζ k V − k , with coeﬃcients V − k . . = e − π i α 2 π i Z 0 −∞  − iΦ( ξ ) ∂ x Φ( ξ ) Φ( ξ ) 2 ( ∂ x Φ( ξ )) 2 iΦ( ξ ) ∂ x Φ( ξ )  ξ k − 1 ∂ s σ Φ ( ξ )d ξ , k ∈ Z > 0 . These coeﬃcients clearly satisfy T r V − k = 0 for every k . Expanding ( 3.42 ) at ζ = ∞ in pow ers of ζ , we obtain a system of equations be t w een the coeﬃcients of B , W 0 and the co eﬃcients V − k . The ﬁrst tw o of suc h non trivial identities read ( [ V − 1 , B 1 ] + [ B 0 , W 0 ] + ∂ s B 0 − ∂ x W 0 = 0 , [ V − 1 , B 0 ] + [ V − 2 , B 1 ] + ∂ x V − 1 = 0 . (3.43) Lo oking at the (1 , 1)-entry of the ﬁrst relation in ( 3.43 ), w e obtain the iden tit y ∂ s p = − i( V − 1 ) 12 = − e − π i α 2 π Z 0 −∞ Φ( ξ ) 2 ∂ s σ Φ ( ξ )d ξ . (3.44) The next step is to integrate this identit y ov er the interv al ( s , ∞ ), and then tak e an x -deriv ative of the result. As a consequence of Proposition 4.8 that will be pro v ed later, p → − 4 α 2 − 1 8 x as s → + ∞ . Also thanks to Prop osition 4.8 , w e kno w that for some function f which is in tegrable in compacts of ( −∞ , 0], gro ws with O ( | ζ | 1 / 4 ) as ζ → −∞ , and is indep enden t of s and x , the bound | Φ( ζ ) | + | ∂ x Φ( ζ ) | ≤ f ( ζ ) is v alid. F rom the explicit form of σ Φ in ( 2.10 ), | ∂ s σ Φ ( ζ ) | = e − s − ( − 1) m ζ m  1 + e − s − ( − 1) m ζ m  2 ≤ e − s − ( − 1) m ζ m , ζ < 0 . Hence, ( | Φ | + | ∂ x Φ | ) ∂ s σ Φ is bounded b y an ( ζ , s )-in tegrable function indep endent of x , and w e can indeed in tegrate ( 3.44 ) in the v ariable s , obtaining the identit y p ( s , x ) = − 4 α 2 − 1 8 x + e − π i α 2 π Z ∞ s Z 0 −∞ Φ( ξ | s = u, x ) 2 ∂ s σ Φ ( ξ | s = u )d ξ d u. (3.45) Diﬀeren tiating no w in x , w e obtain ∂ x p ( s , x ) = 4 α 2 − 1 8 x 2 + e − π i α π Z ∞ s Z 0 −∞ Φ( ξ | s = u, x ) ∂ x Φ( ξ | s = u, x ) ∂ s σ Φ ( ξ | s = u )d ξ d u. Th us, com bining this relation with ( 3.33 ), w e ﬁnally conclude that CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 23 Theorem 3.12. F or ( ζ , s , x ) ∈ R × R × (0 , + ∞ ) , the function Φ = Φ( ζ | s , x ) satisﬁes the nonlo c al PDE ∂ 2 x Φ( ζ | s , x ) =  ζ + 4 α 2 − 1 4 x 2 + 2 e − π i α π Z ∞ s Z 0 −∞ Φ( ξ | u, x ) ∂ x Φ( ξ | u, x ) ∂ s σ Φ ( ξ | u )d ξ d u  Φ( ζ | s , x ) . (3.46) F or reference, we no w obtain a formula that will later be used to relate the limiting kernel for the OP ensem ble ( 2.1 ) with the function Φ just constructed. Using the relation ( 3.21 ), w e obtain the iden tit y h I + e − π i α (1 + e − h ∞ ( ξ ) ) χ S − ( ξ ) E 21  Ψ ∞ ( ξ ) − 1 Ψ ∞ ( ζ )  I − e − π i α (1 + e − h ∞ ( ζ ) ) χ S − ( ζ ) E 21 i 21 , − = h Φ ∞ ( u 1 /m ξ ) − 1 Φ ∞ ( u 1 /m ζ ) i 21 , − , v alid for ζ , ξ ∈ R \ { 0 } . Using ( 3.24 ), ( 3.29 ), ( 3.31 ) and ( 3.32 ), we obtain h I + e − π i α (1 + e − h ∞ ( ξ ) ) χ S − ( ξ ) E 21  Ψ ∞ ( ξ ) − 1 Ψ ∞ ( ζ )  I − e − π i α (1 + e − h ∞ ( ζ ) ) χ S − ( ζ ) E 21 i 21 , − = i e − 2 π i α  Φ(4 ξ / x 2 )( ∂ x Φ)(4 ζ / x 2 ) − Φ(4 ζ / x 2 )( ∂ x Φ)(4 ξ / x 2 )  , where Φ is the same function appearing in ( 3.46 ), and w e used ( 3.34 ). In particular, comparing with ( 2.13 ) w e obtain the RHP representation of the k ernel K α , K α ( ζ , ξ ) = e π i α p σ Φ ( − 4 ξ / x 2 ) p σ Φ ( − 4 ζ / x 2 ) 2 π i( ξ − ζ ) ×  I + e − π i α σ Φ ( − 4 ξ / x 2 ) E 21  Ψ ∞ ( − ξ ) − 1 Ψ ∞ ( − ζ )  I − e − π i α σ Φ ( − 4 ζ / x 2 ) E 21  21 , − , (3.47) whic h is v alid for ξ , ζ ∈ (0 , ∞ ). Recall the notation ( 2.21 ). Sending ξ → ζ , we also obtain the identit y  ∆ ζ  Ψ ∞ ( ζ )  I − e − π i α σ Φ (4 ζ / x 2 ) E 21  21 , − = 2 π i e − π i α σ Φ (4 ζ / x 2 ) K α ( − ζ , − ζ ) , ζ < 0 . (3.48) These representations will b e useful later. W e now mo v e on to computing relev an t asymptotics for the mo del problem. 4. The model RHP: admissible da t a and asymptotics In the course of the analysis of OPs, the mo del problem needed will inv olv e a function h = h τ whic h dep ends on a large parameter τ > 0. Later on, we will set this parameter τ to b e the n um b er of particles n , but during this section w e keep τ as a free parameter. In the limit τ → ∞ , the function h τ will conv erge to h ∞ from ( 3.18 ). The goal of the curren t section is to p erform the required asymptotic analysis of the corresp onding τ -dep endent mo del problem Ψ τ = Ψ ( · | h = h τ ), showing rigorously that it con v erges to the model problem Ψ ∞ = Ψ ∞ ( · | h = h ∞ ) from ( 3.19 ). The remainder of this section is structured as follo ws. In Section 4.1 we in tro duce the conditions on the function h = h τ that w e will b e w orking with. This function h τ will also dep end on the tw o additional parameters s ∈ R and x > 0. In Section 4.2 we analyze the mo del problem Ψ ∞ in the degenerate limit s → + ∞ . This will b e needed in order to in tegrate Ψ ∞ in the v ariable s , to cop e with the limit of the deformation formula ( 2.9 ). This analysis in the limit s → + ∞ will also provide the ma jor steps for the asymptotic analysis when x → 0 + ; the latter will be completed in Section 4.3 . 24 L. MOLAG, G. SIL V A, AND L. ZHANG Finally , in Section 4.4 w e complete the third asymptotic analysis necessary for the model problem, namely the one in the limit τ → + ∞ . 4.1. The class of admissible h . W e now consider the mo del problem Ψ asso ciated to a function h whic h dep ends on a (large) parameter τ in a certain structured manner, as w e introduce next. F or the next statemen t, recall that the con tour Γ w as in troduced in ( 3.1 ) and also Figure 1 . Deﬁnition 4.1. A function h : Γ → C is admissible if it is of the form h ( ζ ) = h τ ( ζ | s ) . . = s + τ 2 m H  ζ τ 2  , where s ∈ R , τ > 0, m ∈ Z > 0 are parameters, and H : Γ → C is a ﬁxed function with the following prop erties. (i) The function H is C ∞ in a neighborho o d of Γ and real-v alued along Γ 0 . (ii) The function H is analytic on a disk D δ with H ( ζ ) = ( − 1) m u ζ m + O ( ζ m +1 ) , ζ → 0 , where u > 0. (iii) There exists Λ > 0 for which Re H ( ζ ) ≥ Λ | ζ | m , ζ ∈ Γ . F or us w e are in terested in the c hoice of parameter τ = n , with n as in ( 2.1 ), but in Deﬁnition 4.1 and for the rest of the presen t section we opt for in tro ducing the new dummy parameter τ , to emphasize that the mo del problem also has an interest on its own. With d s b eing the complex line elemen t in Γ, Deﬁnition 4.1 –(iii) implies, in particular, that | v | k e − H ∈ L 1 (Γ , | d v | ) ∩ L ∞ (Γ , | d v | ) , for an y k > 0 . An y admissible h τ is analytic on a disk cen tered at the origin with growing radius. Moreo v er, the expansion h τ ( ζ ) = s + ( − 1) m u ζ m (1 + O ( τ − 2 )) holds uniformly for ζ in compacts of Γ, with an error term which is indep endent of s . With h ∞ as in ( 3.18 ), this shows in particular that h τ ( ζ ) = h ∞ ( ζ ) + O ( τ − 2 ) , τ → ∞ , (4.1) again with the con v ergence b eing uniform in compacts of Γ, with an error term whic h is indep enden t of s ∈ R . Observ e, in particular, that the righ t-hand side coincides with the data studied in Section 3.3 . Since we are assuming s ∈ R , the con v ergence ab ov e also shows that 1 + e − h is analytic and non- v anishing on a neighborho o d of the origin, with this neigh b orhoo d b eing independent of τ . Therefore, Prop osition 3.2 applies for Ψ with admissible h = h τ , for any τ suﬃciently large, a fact whic h will b e used for the rest of the paper. F rom no w on, we ﬁx h = h τ to alwa ys b e an admissible function in the sense of Deﬁnition 4.1 and denote Ψ τ ( ζ ) . . = Ψ ( ζ | h τ , α ) , Ψ ∞ ( ζ ) . . = Ψ ( ζ | h ∞ , α ) , (4.2) with corresponding jump matrices J τ = J Ψ τ , J ∞ = J Ψ ∞ . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 25 This notation is consistent with ( 3.19 ). The conv ergence ( 4.1 ) indicates that Ψ τ → Ψ ∞ . The main goal of the remainder of this section is to perform the asymptotic analysis to justify this conv ergence, and whic h should b e v alid uniformly for s ≥ s 0 with an y ﬁxed s 0 ∈ R . 4.2. Asymptotic analysis for Ψ ∞ as s → + ∞ . As said, we need to compare Ψ τ with Ψ ∞ with b ounds v alid uniformly for s ∈ [ s 0 , ∞ ), where s 0 ∈ R ﬁxed. In that analysis, we will also require b ounds on appropriate norms of Ψ ∞ v alid uniformly for s ≥ s 0 and also with some rough con trol as ζ → ∞ along Γ. This section is devoted to obtain such b ounds. W e start analyzing Ψ ∞ when s → + ∞ . T o that end, set J (Bes) ( ζ ) . . = ( I + e ± π i α E 21 , ζ ∈ Γ ± , E 12 − E 21 , ζ ∈ Γ 0 . That is, J (Bes) is the jump matrix for the mo del problem obtained w ith data e − h ≡ 0, corresp onding to setting h = + ∞ . Let Ψ (Bes) α b e the solution to the corresp onding RHP , suc h solution is standard and can be constructed explicitly using Bessel functions, see App endix A for more details. The point wise conv ergence J ∞ → J (Bes) , s → + ∞ , is straightforw ard, and leads us to compare Ψ ∞ with Ψ (Bes) α . In fact, for α > − 1 / 2 one could prov e directly that the jump matrix for Ψ ∞ ( Ψ (Bes) α ) − 1 con v erges to the iden tit y matrix in L 2 ∩ L ∞ as s → + ∞ , whic h w ould suﬃce to apply small norm theory to estimate norms of Ψ ∞ and of its boundary v alues. Ho w ever, w e could not impro v e suc h a direct argumen t also to the range − 1 < α ≤ − 1 / 2. The alternate route still in v olv es applying the steep est descent metho d, but the construction of a lo cal parametrix is needed, as we discuss next. Let Ψ 0 = Ψ ∞ , 0 b e the analytic function at the origin obtained when w e apply Prop osition 3.2 to Ψ ∞ , and Ψ 0 = Ψ (Bes) α, 0 the one when we apply the same prop osition to Ψ (Bes) α . F or a suﬃcien tly small ε > 0 whic h w e keep ﬁxed from now on, the function Ψ ∞ , 0 is analytic on D ε . The function λ ( ζ ) . . = 1 2 π i Z 0 −∞  1 1 + e − h ∞ ( s ) − 1  | s | α s − ζ d s, ζ ∈ C \ Γ 0 , (4.3) is analytic on C \ Γ 0 , and satisﬁes λ + ( ζ ) − λ − ( ζ ) =  1 1 + e − h ∞ ( ζ ) − 1  | ζ | α , ζ < 0 . (4.4) Also, from basic properties of Cauc h y transforms, w e obtain the b eha vior λ ( ζ ) =      O ( ζ α ) , if − 1 < α < 0 , O (log ζ ) , if α = 0 , O (1) , if α > 0 , ζ → 0 . (4.5) W e will use λ to construct the lo cal parametrix, and some estimates on λ as s → + ∞ will b e needed. Using the inequality     1 1 + e − h ∞ ( ζ ) − 1     =      e − h ∞ ( ζ ) 1 + e − h ∞ ( ζ )      ≤ e − h ∞ ( ζ ) = e − s e −| ζ | m u , ζ < 0 , 26 L. MOLAG, G. SIL V A, AND L. ZHANG w e obtain that | λ ( ζ ) | ≤ e − s 2 π Z 0 −∞ | s | α e − u | s | m | ζ − s | d s, ζ ∈ C \ Γ 0 . (4.6) Recalling that u > 0, w e learn that λ → 0 uniformly for ζ in compacts of C \ Γ 0 as s → + ∞ . Deforming con tours in the integral deﬁning λ , w e see that this conv ergence also extends uniformly to the whole plane C ; more precisely ∥ λ ∥ L ∞ ( C \ Γ 0 ) = O (e − s ) , ∥ λ ± ∥ L ∞ (Γ 0 ) = O (e − s ) , s → + ∞ . (4.7) W e use this function λ and set P 0 ( ζ ) . . = Ψ (Bes) α, 0 ( ζ ) ( I + λ ( ζ ) E 12 ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 ) × ( I − ( χ S + ( ζ ) e π i α − χ S − ( ζ ) e − π i α )(1 + e − h ∞ ( ζ ) ) E 21 ) , ζ ∈ D ε . (4.8) Next, w e deﬁne R ( ζ ) . . = ( Ψ ∞ ( ζ ) Ψ (Bes) α ( ζ ) − 1 , ζ ∈ C \ (Γ ∪ D ε ) , Ψ ∞ ( ζ ) P 0 ( ζ ) − 1 , ζ ∈ D ε \ Γ . (4.9) It is clear that R is analytic on C \ (Γ ∪ ∂ D ε ). Using the jump prop erties of a and λ giv en in ( 3.8 ) and ( 4.4 ), a cumbersome but straightforw ard calculation sho ws that P 0 , + ( ζ ) = P 0 , − ( ζ ) J ∞ ( ζ ) , ζ ∈ Γ ∩ D ε , that is, P 0 and Ψ ∞ satisfy the same jumps for ζ ∈ Γ ∩ D ε , implying in particular that R has an isolated singularit y at ζ = 0. Also, using ( 4.8 ) and Prop osition 3.2 with Ψ = Ψ ∞ , it follows that R ( ζ ) = Ψ ∞ , 0 ( ζ )(1 + e − h ∞ ( ζ ) ) σ 3 / 2  I +  ζ α a ( ζ )  1 1 + e − h ∞ ( ζ ) − 1  − λ ( ζ )  E 12  Ψ (Bes) α, 0 ( ζ ) − 1 , | ζ | < ε. In virtue of ( 4.5 ), this sho ws that R ( ζ ) = o ( ζ − 1 ) , ζ → 0 , as w ell as R ± ( ζ ) = o ( ζ − 1 ) , ζ → 0 along Γ , (4.10) and therefore the singularit y at ζ = 0 is remov able. In summary , setting Γ R . . = (Γ \ D ε ) ∪ ∂ D ε , where ∂ D ε is orien ted clo ckwise, we conclude that R satisﬁes the following RHP . RHP 4.2. Find a 2 × 2 matrix-v alued function R : C \ Γ R → C 2 × 2 with the following prop erties. (i) R is analytic on C \ Γ R . (ii) The entries of R ± are con tin uous along Γ R , and satisfy the jump relation R + ( ζ ) = R − ( ζ ) J R ( ζ ) with J R ( ζ ) . . =            I + e − h ∞ ( ζ ) e ± π i α Ψ (Bes) α, − ( ζ ) E 21 Ψ (Bes) α, − ( ζ ) − 1 , ζ ∈ Γ ± \ D ε , I − e − h ∞ ( ζ ) 1 + e − h ∞ ( ζ ) Ψ (Bes) α, − ( ζ ) E 11 Ψ (Bes) α, − ( ζ ) − 1 + e − h ∞ ( ζ ) Ψ (Bes) α, − ( ζ ) E 22 Ψ (Bes) α, − ( ζ ) − 1 , ζ ∈ Γ 0 \ D ε , P 0 ( ζ ) Ψ (Bes) α ( ζ ) − 1 , ζ ∈ ∂ D ε , and where we orient ∂ D ε in the clo ckwise direction. (iii) As ζ → ∞ , we ha ve R ( ζ ) = I + O ( ζ − 1 ) . (iv) The en tries of R ± − I belong to L 2 loc (Γ , | d s | ). CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 27 Indeed, the properties (i)–(iii) are immediate from the corresponding prop erties of the RHPs satisﬁed b y Ψ ∞ and Ψ (Bes) α , and property (iv) follows from ( 4.10 ) and the contin uity of P 0 , Ψ (Bes) α , Ψ (Bes) α, − a w ay from the origin. T o conclude the asymptotic analysis, we no w analyze the b ehavior of J R as s → + ∞ . F or the sake of clarit y , w e split this analysis in to the next few lemmas. F or their statements, w e recall that the matrix norm notations were introduced in Section 2 . Lemma 4.3. The estimate ∥ J R − I ∥ L 1 ∩ L ∞ ( ∂ D ε ) = O  e − s  holds as s → + ∞ . Pr o of. Using the deﬁnition of P 0 and Proposition 3.2 with Ψ = Ψ (Bes) α , w e hav e that for ζ ∈ ∂ D ε , J R ( ζ ) = Ψ (Bes) α, 0 ( ζ )( I + λ ( ζ ) E 12 ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 ) × h I − e − h ∞ ( ζ )  χ S + ( ζ ) e απ i − χ S − ( ζ ) e − απ i  E 21 i ( I − a ( ζ ) E 12 ) ζ − α σ 3 / 2 Ψ (Bes) α, 0 ( ζ ) − 1 = I + Ψ (Bes) α, 0 ( ζ ) ζ α σ 3 / 2 h λ ( ζ ) ζ − α E 12 − e − h ∞ ( ζ )  χ S + ( ζ ) e απ i − χ S − ( ζ ) e − απ i  × ( I + ( a ( ζ ) + λ ( ζ ) ζ − α ) E 12 ) E 21 ( I − a ( ζ ) E 12 )  ζ − α σ 3 / 2 Ψ (Bes) α, 0 ( ζ ) − 1 . (4.11) It is clear that e − h ∞ = O (e − s ) uniformly in compacts of C as s → + ∞ . Also, a is indep endent of s . Next, w e already observ ed the con v ergence λ = O (e − s ) as s → + ∞ , uniformly in compacts (see ( 4.7 )). The result then follows once we notice that Ψ (Bes) α, 0 and its in verse are b oth indep enden t of s and con tin uous functions of ζ , and the contour ∂ D ε is ob viously b ounded. □ Lemma 4.4. The estimate ∥ J R − I ∥ L 1 ∩ L ∞ (Γ + ∪ Γ − \ D ε ) = O  e − s  holds as s → + ∞ . Pr o of. The asymptotic behavior in ( 3.5 ) for Ψ = Ψ (Bes) α giv es that as ζ → ∞ along Γ + ∪ Γ − , Ψ (Bes) α, − ( ζ ) E 21 Ψ (Bes) α, − ( ζ ) − 1 = O (e − 4 ζ 1 / 2 ζ 1 / 2 ) , with all the terms in the identit y ab ov e b eing indep endent of s . Along Γ − ∪ Γ + , we hav e Re ζ 1 / 2 = | ζ | 1 / 2 cos θ m 2 > 0, with θ m as in ( 3.1 ), and therefore the term on the right-hand side ab ov e has ﬁnite L 1 and L ∞ norms along Γ + ∪ Γ − . Likewise, the function e ( − 1) m +1 u ζ m = e − h ∞ ( ζ )+ s has ﬁnite L 1 and L ∞ norms along Γ + ∪ Γ − , whic h are indep enden t of s as well. F rom the inequality ∥ J R − I ∥ L 1 ∩ L ∞ (Γ + ∪ Γ − ) ≤ e − s ∥ e − h ∞ + s ± π i α Ψ (Bes) α, − E 21 ( Ψ (Bes) α, − ) − 1 ∥ L 1 ∩ L ∞ (Γ + ∪ Γ − ) , the lemma then follo ws. □ Lemma 4.5. The estimate ∥ J R − I ∥ L 1 ∩ L ∞ (Γ 0 \ D ε ) = O  e − s  holds as s → + ∞ . 28 L. MOLAG, G. SIL V A, AND L. ZHANG Pr o of. Using again the asymptotic b ehavior in ( 3.5 ) for Ψ = Ψ (Bes) α , w e obtain that as ζ → ∞ along Γ 0 , and for j = 1 , 2 e − h ∞ ( ζ )+ s Ψ (Bes) α, − ( ζ ) E j j Ψ (Bes) α, − ( ζ ) − 1 = O (e − u | ζ | m ζ 1 / 2 ) , (4.12) with all the terms in the identit y ab ov e b eing independent of s . Using that h ∞ is real along the real axis, w e obtain the inequality | 1 / (1 + e − h ∞ ) | ≤ 1 for ζ < 0. Pro ceeding now as in the proof of Lemma 4.4 , these estimates are suﬃcien t to conclude the pro of. □ W e ﬁnally conclude Prop osition 4.6. The estimates ∥ R − I ∥ L ∞ ( C \ Γ R ) = O (e − s ) , ∥ R ± − I ∥ L 1 ∩ L 2 ∩ L ∞ (Γ R ) = O (e − s ) ar e valid as s → + ∞ . Pr o of. The estimates on L 1 ∩ L ∞ from Lemmas 4.3 – 4.5 extend to estimates v alid on L 2 ∩ L ∞ as w ell, and the current lemma then follows from standard p erturbation theory for RHPs. □ F or later, we will need some consequences of the asymptotic analysis just carried out. W e start stating a rough estimate for Ψ ∞ , v alid uniformly in s . Prop osition 4.7. Fix s 0 ∈ R . Ther e exist c onstants M > 0 and η > 0 such that the ine qualities   Ψ ∞ , − ( ζ ) E 21 Ψ ∞ , − ( ζ ) − 1   ≤ M e − η | ζ | 1 / 2 , ζ ∈ (Γ + ∪ Γ − ) \ D ε , and   Ψ ∞ , − ( ζ ) E kk Ψ ∞ , − ( ζ ) − 1   ≤ M | ζ | 1 / 2 , ζ ∈ Γ 0 \ D ε , k = 1 , 2 , ar e valid for any s ≥ s 0 . Pr o of. Since the jumps of R are analytic, standard arguments on the asymptotic analysis of R as s → + ∞ ensure that Ψ ∞ , − ( ζ ) = ( I + O (e − s )) Ψ (Bes) α, − ( ζ ) , uniformly along Γ \ D ε as s → + ∞ . The result no w follows from the asymptotics ( 3.5 ) for Ψ = Ψ (Bes) α ( ζ ) and the fact that Ψ (Bes) α, − is con tin uous along Γ \ D ε and hence b ounded on compact subsets of Γ \ D ε . □ Next, w e obtain estimates for quantities related to the in tegrable systems studied in Section 3.3 . It is w orth mentioning that the estimate in (ii) b elow is not optimal, but suﬃcient for our purposes. Prop osition 4.8. With the functions p = p ( s , x ) and Φ( ζ ) = Φ( ζ | s , x ) intr o duc e d in ( 3.26 ) and ( 3.32 ) , r esp e ctively, we have the fol lowing estimates. (i) The estimate p ( s , x ) = − 4 α 2 − 1 8 x + O (e − s ) is valid as s → + ∞ , uniformly for x > 0 in c omp acts of (0 , + ∞ ) . (ii) Ther e exists δ > 0 and M > 0 such that the estimate | Φ( ζ | s , x ) | + | ∂ x Φ( ζ | s , x ) | ≤ M  | ζ | 1 / 4 χ ( −∞ , − δ ) ( ζ ) + ( δ 0 ( α )(log | ζ | ) 2 + | ζ | −| α | / 2 ) χ ( − δ, 0) ( ζ )  , ζ < 0 , is valid as s → + ∞ , uniformly for x > 0 in c omp acts of (0 , + ∞ ) . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 29 Pr o of. The RHP 4.2 for R is equiv alen t to the follo wing in tegral equation: R ( ζ ) = I + 1 2 π i Z Γ R R − ( s )( J R ( s ) − I ) d s s − ζ , ζ ∈ C \ Γ R . F rom this identit y , condition (iii) in RHP 4.2 ma y b e impro ved to R ( ζ ) = I + 1 ζ R 1 + O ( ζ − 2 ) , ζ → ∞ , with R 1 . . = − 1 2 π i Z Γ R R − ( s )( J R ( s ) − I )d s. Thanks to Prop osition 4.6 , R 1 = O (e − s ) as s → + ∞ . On the other hand, from ( 4.9 ) and ( A.10 ) we get the relation Ψ ∞ , 1 = R 1 + Ψ (Bes) ∞ , 1 ( α ) . (4.13) Com bining with ( 3.26 ) and ( A.11 ), the result ab out p follows. The result about Φ follo ws in a similar manner, as we no w outline. The claimed estimate for ζ < 0 and a w a y from the origin follows from ( 3.36 ). F rom relations ( 3.21 ), ( 3.32 ) and ( 3.24 ), it follows that Φ(4 ζ / x 2 ) =  x 2  1 / 2 h Ψ ∞ ( ζ )  I + χ S + ( ζ ) e π i α (1 + e − h ∞ ( ζ ) ) E 21 i 11 =       x 2  1 / 2 ( Ψ ∞ ) 11 ( ζ ) , ζ > 0 ,  x 2  1 / 2 h ( Ψ ∞ ) 11 ( ζ ) + e π i α (1 + e − h ∞ ( ζ ) ) ( Ψ ∞ ) 12 ( ζ ) i , ζ < 0 . (4.14) F or − ε < ζ < 0, w e use ( 4.8 ) and ( 4.9 ), together with Prop osition 4.6 , to express Ψ ∞ , + ( ζ ) =  I + O (e − s )  Ψ (Bes) α, 0 ( ζ )( I + λ + ( ζ ) E 12 ) ζ α σ 3 / 2 + ( I + a + ( ζ ) E 12 )( I − e π i α (1 + e − h ∞ ( ζ ) ) E 21 ) . The term Ψ (Bes) α, 0 is indep endent of s and analytic in ζ , hence b ounded for ζ in a neighborho o d of the origin. F rom ( 3.7 ), a + ( ζ ) = O (1 + δ 0 ( α ) log | ζ | ). F rom ( 4.5 ), we also learn that ( I + λ + ( ζ ) E 12 ) ζ α σ 3 / 2 + = O ( | ζ | −| α | / 2 + δ 0 ( α ) log | ζ | ). Finally , from ( 3.18 ), w e see that e − h ∞ ( ζ ) = O (e − s ) as s → + ∞ , uniformly for ζ < 0. The result now follows from ( 4.14 ). □ 4.3. Asymptotic analysis of Ψ ∞ as u → + ∞ . W e also need to p erform the asymptotic analysis of Ψ ∞ as u → + ∞ . The latter limit corresp onds to x → 0 + , and this asymptotic analysis will pro vide (singular) b oundary conditions for the functions p and Φ. Our starting p oint in this asymptotic analysis will already b e the function R from ( 4.9 ), which is the solution to RHP 4.2 . The remaining c hallenge here is the analysis of the jump for R as u → + ∞ , as opp osed to s → + ∞ provided in the previous section. In other words, we need the analogues of Lemmas 4.3 – 4.5 , which we no w pro vide. Lemma 4.9. The estimate ∥ J R − I ∥ L 1 ∩ L ∞ ( ∂ D ε ) = O ( u − (1+ α ) /m ) , u → + ∞ , holds uniformly for s ≥ s 0 with ﬁxe d s 0 ∈ R . 30 L. MOLAG, G. SIL V A, AND L. ZHANG Pr o of. W e analyze J R along ∂ D ε through its explicit expression from ( 4.11 ), which for conv enience of the reader we recall to b e J R ( ζ ) = I + Ψ (Bes) α, 0 ( ζ ) ζ α σ 3 / 2 h λ ( ζ ) ζ − α E 12 − e − h ∞ ( ζ )  χ S + ( ζ ) e απ i − χ S − ( ζ ) e − απ i  × ( I + ( a ( ζ ) + λ ( ζ ) ζ − α ) E 12 ) E 21 ( I − a ( ζ ) E 12 )  ζ − α σ 3 / 2 Ψ (Bes) α, 0 ( ζ ) − 1 . W e next estimate the righ t-hand side ab o v e term by term. The function Ψ (Bes) α, 0 and its matrix inv erse ( Ψ (Bes) α, 0 ) − 1 are analytic in ζ and indep enden t of u , hence remain uniformly b ounded for ζ ∈ ∂ D ε . The inequality ( 4.6 ), together with a straigh tforw ard asymp- totic analysis of the right-hand side through the classical saddle p oint metho d, unra vels the rough estimate λ ( ζ ) = O  u − (1+ α ) /m  , u → + ∞ , v alid uniformly for s ≥ s 0 , and uniformly for ζ in compacts of C \ Γ 0 . A standard argument using deformation of con tours shows that the same estimate is also v alid do wn to the b oundary v alues λ ± ( ζ ) for ζ along Γ 0 , and therefore it is v alid uniformly for ζ ∈ ∂ D ε . The factors ζ α σ 3 / 2 and a are contin uous functions of ζ ∈ C \ Γ 0 , with con tin uous boundary v alues along Γ 0 , and indep enden t of u . Hence, they remain bounded for ζ ∈ ∂ D ε . Finally , from the explicit form of h ∞ from ( 3.18 ) and ( 3.3 ), we see that | e − h ∞ ( ζ ) | ≤ e − s − u | ζ | m / 2 , ζ ∈ S + ∪ S − . Hence, for some η > 0, the estimate e − h ( ζ ) χ S ± ( ζ ) = O (e − η u ) holds uniformly for ζ ∈ ∂ D ε . Plugging in all these estimates in to ( 4.11 ), w e obtain the lemma in the L ∞ norm, and the L 1 claim follo ws from it and compactness of ∂ D ε . □ R emark 4.10 . The proof of Lemma 4.9 actually sho ws a sligh tly stronger result, namely that J R ( ζ ) = I + λ ( ζ ) Ψ (Bes) α, 0 ( ζ ) E 12 Ψ (Bes) α, 0 ( ζ ) − 1 + O (e − η u ) , u → ∞ , uniformly for ζ ∈ ∂ D ε . Performing the c hange of v ariables u 1 /m s = v in ( 4.3 ), we obtain the asymptotic expansion λ ( ζ ) ∼ ∞ X k =0 λ k i u α +1+ k m ζ k +1 , λ k = λ k ( s ) . . = ( − 1) k 2 π Z 0 −∞ | v | α + k e − s − ( − 1) m v m 1 + e − s − ( − 1) m v m d v , k ∈ Z ≥ 0 . (4.15) This, together with the analyticity of Ψ (Bes) α, 0 near the origin, allows us to compute an asymptotic expansion of J R of the form J R ( ζ ) ∼ I + ∞ X k =1 1 u α + k m J k ( ζ ) , u → + ∞ , with J k ( ζ ) . . = λ k − 1 i ζ k Ψ (Bes) α, 0 ( ζ ) E 12 Ψ (Bes) α, 0 ( ζ ) − 1 , (4.16) v alid uniformly for ζ ∈ ∂ D ε . Notice that J k has a p ole of order at most k at ζ = 0. ▷ Lemma 4.11. Given s 0 ∈ R , ther e exists η > 0 such that the estimate ∥ J R − I ∥ L 1 ∩ L ∞ ((Γ + ∪ Γ − ) \ D ε ) = O (e − η u ) , u → + ∞ , is valid uniformly for s ≥ s 0 . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 31 Pr o of. Pro ceeding as in the proof of Lemma 4.4 , w e see that ∥ J R − I ∥ L 1 ∪ L ∞ ((Γ + ∩ Γ − ) \ D ε ) ≤ ∥ Ψ (Bes) α, − E 21 ( Ψ (Bes) α, − ) − 1 ∥ L ∞ ((Γ + ∪ Γ − ) \ D ε ) ∥ e − h ∞ ∥ L 1 ∩ L ∞ ((Γ + ∪ Γ − ) \ D ε ) , and the norm of Ψ (Bes) α, − E 21 ( Ψ (Bes) α, − ) − 1 ab o v e is ﬁnite. F rom ( 3.3 ), the norms of e − h ∞ ab o v e are O (e − η u ), and the result follo ws. □ Lemma 4.12. Given s 0 ∈ R , ther e exists η > 0 such that the estimate ∥ J R − I ∥ L 1 ∩ L ∞ (Γ 0 \ D ε ) = O (e − η u ) , u → + ∞ , is valid uniformly for s ≥ s 0 . Pr o of. F rom the very deﬁnition of J R in RHP 4.2 , it follows that, for ζ ∈ Γ 0 \ D ε ), | J R ( ζ ) − I | ≤ | e − h ∞ ( ζ ) |  | Ψ (Bes) α, − ( ζ ) E 11 Ψ (Bes) α, − ( ζ ) − 1 | + | Ψ (Bes) α, − ( ζ ) E 22 Ψ (Bes) α, − ( ζ ) − 1 |  . The result now follows from ( 4.12 ). □ The asymptotic analysis is concluded with the next result. Prop osition 4.13. The estimates ∥ R − I ∥ L ∞ ( C \ Γ R ) = O ( u − (1+ α ) /m ) and ∥ R ± − I ∥ L 1 ∩ L 2 ∩ L ∞ (Γ R ) = O ( u − (1+ α ) /m ) ar e valid as u → + ∞ , uniformly for s ≥ s 0 with ﬁxe d s 0 ∈ R . Pr o of. With the help of Lemmas 4.9 , 4.11 and 4.12 , the conclusion is straightforw ard from perturbation theory for RHPs, and we skip it. □ W e no w dra w the conclusions from this asymptotic analysis, namely asymptotic form ulas for the function p and Φ from ( 3.26 ) and ( 3.32 ). F or the next statement and its pro of, recall that x = 2 u − 1 2 m , see ( 3.24 ). Prop osition 4.14. With λ 0 as in ( 4.15 ) , the estimate p ( s , x ) = − 4 α 2 − 1 8 x + 2 π λ 0 2 2 α +1 Γ( α + 1) 2 x 2 α +1 + O ( x 2 α +3 ) , x → 0 + , holds. In addition, the asymptotic formula Φ( ζ | s , x ) = √ π x 1 / 2 I α ( x ζ 1 / 2 )  1 + O ( x 2+2 α (1 + x | ζ | 1 / 2 ))  , ζ ∈ R \ { 0 } , (4.17) is valid as x → 0 + , uniformly for ζ ∈ R \ { 0 } , and uniformly for s ≥ s 0 with ﬁxe d s 0 ∈ R . R emark 4.15 . Ev en though ( 4.17 ) is v alid for ζ ∈ R \ { 0 } , it b ecomes useful only when | ζ | ≪ x − (4 α +2) , as in this case the O ( · ) term is truly a small error term. Nev ertheless, this is a genuine asymptotic form ula, in particular, for ζ in compacts of R . ▷ 32 L. MOLAG, G. SIL V A, AND L. ZHANG Pr o of. F rom ( 3.26 ), ( 4.13 ) and ( A.11 ), it follo ws that p ( s , x ) = − 4 α 2 − 1 8 x + 2i x ( R 1 ) 12 , (4.18) where w e recall that R 1 = − 1 2 π i Z Γ R R − ( s )( J R ( s ) − I )d s = − 1 2 π i Z ∂ D ε R − ( s )( J R ( s ) − I )d s + O (e − η u ) , and the last equalit y is a consequence of Prop osition 4.13 and Lemmas 4.11 and 4.12 . W riting R − = ( R − − I ) + I , and app ealing again to Prop osition 4.13 and Remark 4.10 , we get R 1 = − x 2 α +2 2 2 α +3 π i Z ∂ D ε J 1 ( s )d s + O ( x 4 α +4 ) , x → 0 + , with J 1 as in ( 4.16 ). This c oeﬃcient has a simple p ole at ζ = 0 and no other p oles, so a residue calculation pro vides Z ∂ D ε J 1 ( s )d s = − 2 π λ 0 Ψ (Bes) α, 0 (0) E 12 Ψ (Bes) α, 0 (0) − 1 , with λ 0 giv en in ( 4.15 ). W e update ( 4.18 ) to p ( s , x ) = − 4 α 2 − 1 8 x + λ 0 2 2 α +1 x 2 α +1 h Ψ (Bes) α, 0 (0) E 12 Ψ (Bes) α, 0 (0) − 1 i 12 + O ( x 4 α +3 ) = − 4 α 2 − 1 8 x + λ 0 2 2 α +1 x 2 α +1 h Ψ (Bes) α, 0 (0) i 11  2 + O ( x 4 α +3 ) = − 4 α 2 − 1 8 x + 2 π λ 0 2 2 α +1 Γ( α + 1) 2 x 2 α +1 + O ( x 4 α +3 ) , x → 0 + , where w e hav e made use of ( A.8 ) and ( A.9 ) in the last equalit y . Next, for the asymptotics of Φ, w e ﬁrst use ( 4.14 ) and ( 4.9 ) to express Φ( ζ ) = x 1 / 2 √ 2 ×    h R ( x 2 ζ / 4) Ψ (Bes) α ( x 2 ζ / 4) i 11 , ζ > 4 x 2 ε ( x 2 ζ / 4) α/ 2 h R ( x 2 ζ / 4) Ψ (Bes) α, 0 ( x 2 ζ / 4) i 11 , 0 < ζ < 4 x 2 ε. The term in the second line ab ov e dep ends solely on the ﬁrst column of Ψ (Bes) α, 0 . In view of ( A.6 ), ( A.7 ) and ( A.9 ), w e are then able to re-express this iden tit y in an y of the tw o equiv alent and uniform w a ys, Φ( ζ ) = x 1 / 2 √ 2 ( x 2 ζ / 4) α/ 2 h R ( x 2 ζ / 4) Ψ (Bes) α, 0 ( x 2 ζ / 4) i 11 = x 1 / 2 √ 2 h R ( x 2 ζ / 4) Ψ (Bes) α ( x 2 ζ / 4) i 11 , ζ > 0 . (4.19) Applying Proposition 4.13 , we obtain Φ( ζ ) = x 1 / 2 √ 2 h  I + O ( x 2+2 α )  Ψ (Bes) α ( x 2 ζ / 4) i 11 = x 1 / 2 √ 2  Ψ (Bes) α ( x 2 ζ / 4)  11 1 + O ( x 2+2 α ) + [ Ψ (Bes) α ( x 2 ζ / 4)] 21 [ Ψ (Bes) α ( x 2 ζ / 4)] 11 O ( x 2+2 α ) ! , v alid as x → 0 + , uniformly for ζ > 0. Giv en a suﬃcien tly large M > , we see from A that the quotien t ( Ψ (Bes) α ) 21 / ( Ψ (Bes) α ) 11 is O ( ζ 1 / 2 x ) for x 2 ζ ≥ M . On the other hand, this same quotient is an en tire CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 33 function of ζ (see ( A.1 ), ( A.9 ) and ( A.5 )), and therefore it is O (1) for 0 ≤ ζ x 2 ≤ M . In summary , we simplify this last asymptotic identit y to Φ( ζ ) = x 1 / 2 √ 2  Ψ (Bes) α ( x 2 ζ / 4)  11  1 + O ( x 2+2 α (1 + x | ζ | 1 / 2 ))  , x → 0 + . The asymptotic claim for ζ > 0 no w follo ws from ( A.9 ) and ( A.1 ). The claim for ζ < 0 is very similar: a cumbersome but direct calculation sho ws that the ﬁrst identit y in ( 4.19 ) still holds for ζ < 0, and the remaining of the argumen ts remain pretty muc h the same. □ 4.4. Asymptotic analysis of the mo del RHP with admissible data. Recall that the matrix-v alued function Ψ = Ψ τ w as in tro duced in ( 4.2 ) as the solution to the mo del problem RHP 3.1 , with admis sible data h = h τ as introduced in Deﬁnition 4.1 . As highligh ted in ( 4.1 ) et se q. , we exp ect that Ψ τ → Ψ ∞ as τ → + ∞ , with the latter b eing the solution to the RHP with data h ∞ giv en in ( 3.18 ), and the goal of this section is to sho w such conv ergence. The asymptotic analysis is very similar to the one carried out in the previous section. The ﬁrst step is the construction of a lo cal parametrix P τ in a ﬁxed disk D ε near the origin, whic h is the solution to the follo wing RHP . RHP 4.16. Find a 2 × 2 matrix-v alued function P τ : D ε \ Γ → C 2 × 2 with the following prop erties. (i) P τ is analytic on D ε \ Γ. (ii) On Γ ∩ D ε \ { 0 } , P τ satisﬁes the jump relation P τ , + ( ζ ) = P τ , − ( ζ ) J τ ( ζ ) , where w e recall that J τ ( ζ ) =    I + (1 + e − h τ ( ζ ) ) e ± π i α E 21 , ζ ∈ Γ ± , 1 1 + e − h τ ( ζ ) E 12 − (1 + e h τ ( ζ ) ) E 21 , ζ ∈ Γ 0 , with h τ as in Deﬁnition 4.1 . (iii) As τ → ∞ , P τ ( ζ ) = ( I + O (e − s τ − 2 )) Ψ ∞ ( ζ ) , uniformly for ζ ∈ ∂ D ε , and also uniformly for s ≥ s 0 with an y ﬁxed s 0 ∈ R . T o construct the solution to this RHP , we ﬁrst introduce the analogue of ( 4.3 ) to our context here, namely the function λ τ ( ζ ) . . = 1 2 π i Z 0 −∞  1 1 + e − h τ ( s ) − 1  | s | α s − ζ d s, ζ ∈ C \ Γ 0 . (4.20) Using that h τ is real-v alued along the negative real axis, we obtain the inequalit y     1 1 + e − h τ ( ζ ) − 1     ≤ e − h τ ( ζ ) , ζ < 0 . (4.21) Com bined with the in tegrabilit y condition ask ed in Deﬁnition 4.1 –(iii), w e are ensured that λ τ is indeed w ell-deﬁned for ζ ∈ C \ Γ 0 . In fact it is the Cauc h y transform of a function in L 1 ∩ L ∞ (Γ 0 ), th us λ τ , ± b elongs to L p for any p ∈ (1 , ∞ ). Since h τ is analytic on a disk growing with τ , we also see that for any compact K ⊂ Γ 0 ﬁxed, there exists τ 0 = τ 0 ( K ) for which λ τ , ± ∈ L ∞ ( K ) for any τ ≥ τ 0 . This latter claim follo ws from deformation of con tours in the deﬁnition of λ τ . 34 L. MOLAG, G. SIL V A, AND L. ZHANG Fix a compact K ⊂ C \ { 0 } . Using the inequality ( 4.21 ) and again the analyticit y of h τ in a τ -indep endent neighborho o d of the real axis, standard arguments sho w that there exists a constan t C K > 0, dep ending solely on K , for which | λ τ ( ζ ) | ≤ C K Z 0 −∞ | s | α e − h τ ( ζ ) d s, for ev ery ζ ∈ K . Similarly as for λ , the function λ τ satisﬁes the jump condition λ τ , + ( ζ ) − λ τ , − ( ζ ) =  1 1 + e − h τ ( ζ ) − 1  | ζ | α = − | ζ | α e − h τ ( ζ ) 1 + e − h τ ( ζ ) , ζ < 0 , (4.22) and the b ehavior λ τ ( ζ ) =      O ( ζ α ) , − 1 < α < 0 , O (log ζ ) , α = 0 , O (1) , α > 0 , (4.23) as ζ → 0. Using the function λ τ , w e construct a solution to RHP 4.16 in the form P τ ( ζ ) . . = E τ ( ζ ) ( I + λ τ ( ζ ) E 12 ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 ) × h I − ( χ S + ( ζ ) e π i α − χ S − ( ζ ) e − π i α )(1 + e − h τ ( ζ ) ) E 21 i , ζ ∈ D ε \ Γ , (4.24) where we recall that χ S ± is the c haracteristic function of the set S ± previously introduced in ( 3.2 ), the function a is as in ( 3.7 ), and with the term Ψ ∞ , 0 = Ψ 0 b eing the matrix-v alued analytic function obtained when we apply Proposition 3.2 to Ψ = Ψ ∞ , w e used the matrix prefactor E τ ( ζ ) . . = Ψ ∞ , 0 ( ζ )  1 + e − h ∞ ( ζ )  σ 3 / 2 I − e − h τ ( ζ ) a ( ζ ) ζ α 1 + e − h τ ( ζ ) + λ τ ( ζ ) ! E 12 ! , ζ ∈ D ε . (4.25) W e no w v erify that ( 4.24 ) is indeed a solution to RHP 4.16 . First of all, the factor E τ is analytic near the origin. Indeed, all the terms inv olved in the deﬁnition of E τ are analytic except across the negativ e axis. F or ζ < 0, w e compute E τ , − ( ζ ) − 1 E τ , + ( ζ ) = I − λ τ , + ( ζ ) − λ τ , − ( ζ ) + e − h τ ( ζ ) 1 + e − h τ ( ζ )  a + ( ζ ) e π i α − a − ( ζ ) e − π i α  | ζ | α ! E 12 , and using ( 4.22 ) and ( 3.8 ) the right-hand side ab ov e simpliﬁes to I . This shows that E τ is analytic across the negative axis as w ell, so it has an isolated singularity at the origin. F rom the deﬁnition of E τ , ( 3.7 ) and ( 4.23 ), we see that E τ ( ζ ) = o ( ζ − 1 ) as ζ → 0, so this singularit y is in fact remov able. W e now come bac k to P τ . Every factor in the righ t-hand side of ( 4.24 ) is analytic on C \ Γ, so the same holds for P τ , that is, RHP 4.16 –(i) is satisﬁed. All the factors on the ﬁrst line of the righ t-hand side of ( 4.24 ) are analytic across Γ ± , and the v eriﬁcation of RHP 4.16 –(ii) on Γ ± is immediate. After a cum b ersome but straigh tforw ard calculation using ( 4.22 ) and ( 3.7 ), we verify that RHP 4.16 –(ii) is also satisﬁed along Γ 0 . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 35 Finally , we no w v erify RHP 4.16 –(iii). Using the deﬁnition of E τ and Prop osition 3.2 applied to Ψ ∞ , w e express for ζ ∈ ∂ D ε , P τ ( ζ ) Ψ ∞ ( ζ ) − 1 = Ψ ∞ , 0 ( ζ ) ζ α 2 σ 3 ( I + a ( ζ ) E 12 ) (1 + e − h ∞ ( ζ ) ) σ 3 / 2 ×  I −  1 1 + e − h τ ( ζ ) − 1 1 + e − h ∞ ( ζ )  a ( ζ ) E 12   I −  χ S + ( ζ ) e π i α − χ S − ( ζ ) e − π i α  (e − h τ ( ζ ) − e − h ∞ ( ζ ) ) E 21  × (1 + e − h ∞ ( ζ ) ) − σ 3 / 2 ( I − a ( ζ ) E 12 ) ζ − α 2 σ 3 Ψ ∞ , 0 ( ζ ) − 1 . (4.26) The term on the last line of ( 4.26 ) is the inv erse of the term on the righ t-hand side of the ﬁrst line, and these terms are indep enden t of τ . F rom the explicit expression for h ∞ (recall ( 4.1 )), it follows that w e can c ho ose ε > 0 suﬃcien tly small and independent of s ∈ R in such a wa y that suc h poles remain within a p ositive distance from D ε . As suc h, the fraction (1 + e − h ∞ ( ζ ) ) − 1 is bounded on D ε uniformly for s ≥ s 0 with an y ﬁxed s 0 ∈ R . All in all, it means that the terms on the ﬁrst and last lines of ( 4.26 ) are b ounded functions of ζ ∈ ∂ D ε , uniformly in s ≥ s 0 . Hence, to conclude RHP 4.16 –(iii) it suﬃces to pro v e that the terms on the second line of ( 4.26 ) are of the form I + O (e − s τ − 2 ). F rom the conv ergence in ( 4.1 ), w e hav e e − h τ ( ζ ) − e − h ∞ ( ζ ) = e − h ∞ ( ζ ) (1 − e h ∞ ( ζ ) − h τ ( ζ ) ) = O ( τ − 2 e − s ) , (4.27) as τ → ∞ , uniformly for ζ in compacts (and in particular for ζ ∈ ∂ D ε ) and also uniformly for s ≥ s 0 with an y ﬁxed s 0 ∈ R . This estimate tak es care of the second term in the second line of ( 4.26 ). W e already observed that (1 + e − h ∞ ( ζ ) ) − 1 remains uniformly b ounded for ζ ∈ ∂ D ε , and from the uniform conv ergence ( 4.1 ) we learn that (1 + e − h τ ( ζ ) ) − 1 remains to o uniformly b ounded for ζ ∈ D ε and s ∈ R , and also uniformly as τ → + ∞ . This w a y , w e no w b ound     1 1 + e − h τ ( ζ ) − 1 1 + e − h ∞ ( ζ )     = | e − h τ ( ζ ) − e − h ∞ ( ζ ) | | 1 + e − h τ ( ζ ) || 1 + e − h ∞ ( ζ ) | = O (e − s τ − 2 ) , τ → ∞ , (4.28) uniformly for ζ in compacts (and again in particular for ζ ∈ ∂ D ε ), where we used again ( 4.27 ). This is the required estimate for the ﬁrst term in the second line of ( 4.26 ), and concludes the pro of that P τ from ( 4.24 ) solves RHP 4.16 –(iii). T o complete the asymptotic analysis, we use transform R ( ζ ) . . = ( Ψ τ ( ζ ) Ψ ∞ ( ζ ) − 1 , ζ ∈ C \ (Γ ∪ D ε ) , Ψ τ ( ζ ) P τ ( ζ ) − 1 , ζ ∈ D ε \ Γ . (4.29) Recall that Γ R . . = Γ ∪ ∂ D ε \ D ε , w e ha v e that R satisﬁes the follo wing RHP . RHP 4.17. Find a 2 × 2 matrix-v alued function R : C \ Γ R → C 2 × 2 with the following prop erties. (i) R is analytic on C \ Γ R . (ii) The entries of R ± are con tin uous along Γ R , and satisfy the jump relation R + ( ζ ) = R − ( ζ ) J R ( ζ ) with J R ( ζ ) . . =            I +  e − h τ ( ζ ) − e − h ∞ ( ζ )  e ± π i α Ψ ∞ , − ( ζ ) E 21 Ψ ∞ , − ( ζ ) − 1 , ζ ∈ Γ ± \ D ε , I +  e − h ∞ ( ζ ) − e − h τ ( ζ )  Ψ ∞ , − ( ζ )  E 11 1 + e − h τ ( ζ ) − E 22 1 + e − h ∞ ( ζ )  Ψ ∞ , − ( ζ ) − 1 , ζ ∈ Γ 0 \ D ε , P τ ( ζ ) Ψ ∞ ( ζ ) − 1 , ζ ∈ ∂ D ε , 36 L. MOLAG, G. SIL V A, AND L. ZHANG and where we orient ∂ D ε in the clo ckwise direction. (iii) As ζ → ∞ , we ha ve R ( ζ ) = I + O  ζ − 1  . Next, w e estimate the jumps of R to conclude the asymptotic analysis. Lemma 4.18. Fix s 0 ∈ R . F or any κ ∈ (0 , 2) , the estimate ∥ J R − I ∥ L 1 ∩ L ∞ (Γ R ) = O (e − s τ − κ ) holds uniformly for s ≥ s 0 as τ → ∞ . Pr o of. F or a giv en κ ∈ (0 , 2), set ν . . = (2 − κ ) / ( m + 1) ∈ (0 , 2 / ( m + 1)). W e will make the estimates in three separate subsets of Γ R , namely on ∂ D ε , (Γ R ∩ D τ ν ) \ ∂ D ε and Γ R \ D τ ν . F rom RHP 4.16 –(iii) w e immediately obtain ∥ J R − I ∥ L 1 ∩ L ∞ ( ∂ D ε ) = O (e − s τ − 2 ) . (4.30) Next, we mov e to the required estimates on Γ R ∩ D τ ν \ ∂ D ε . W e learn from ( 4.1 ) and Deﬁnition 4.1 –(ii) that there exists M > 0, indep enden t of s , τ , for whic h | h ∞ ( ζ ) − h τ ( ζ ) | ≤ M τ 2 − ( m +1) ν = M τ κ , ζ ∈ D τ ν . In particular h τ − h ∞ → 0 uniformly for ζ ∈ D τ ν as τ → ∞ . F rom the inequality | 1 − e w | ≤ | w | e | w | , v alid for any w ∈ C , and the fact that | e − h ∞ ( ζ )+ s | is independent of s and b ounded b y e − η | ζ | m along Γ for some η > 0, w e learn | e − h ∞ ( ζ ) − e − h τ ( ζ ) | ≤ | e − h ∞ ( ζ ) || h ∞ ( ζ ) − h τ ( ζ ) | e | h ∞ ( ζ ) − h τ ( ζ ) | ≤ M e − s e − η | ζ | m τ κ , ζ ∈ Γ ∩ D τ ν , (4.31) for a p ossibly diﬀerent constant M > 0, but still indep enden t of s , τ . Because h τ , h ∞ are real-v alued along ( −∞ , 0), we ha ve | 1 + e − h τ | − 1 , | 1 + e − h ∞ | − 1 ≤ 1. Then, using ( 4.31 ) and Prop osition 4.7 , w e conclude from J R in RHP 4.17 –(ii) that ∥ J R − I ∥ L 1 ∩ L ∞ ((Γ R ∩ D τ ν ) \ ∂ D ε ) ≤ M e − s τ κ ∥ ζ 1 / 2 e − η | ζ | m ∥ L 1 ∩ L ∞ ((Γ R ∩ D τ ν ) \ ∂ D ε ) = O (e − s τ − κ ) . (4.32) Finally , w e mov e to obtaining the appropriate b ounds on Γ R \ D τ ν . The very deﬁnition of h ∞ in ( 3.18 ) already shows that for some η 1 > 0, | e − h ∞ ( ζ ) | ≤ e − s e − 2 η 1 | ζ | m ≤ e − s − η 1 τ mν e − η 1 | ζ | m , ζ ∈ Γ R \ D τ ν . Next, from Deﬁnition 4.1 –(iii) we learn that | e − h τ ( ζ ) | ≤ e − s e − Λ τ ν / 2 e − Λ | ζ | ν / 2 , ζ ∈ Γ R \ D τ ν . Com bining these tw o inequalities, w e obtain the estimate | e − h τ ( ζ ) − e − h ∞ ( ζ ) | ≤ M e − s − η τ ν − η | ζ | ν , ζ ∈ Γ R \ D τ ν (4.33) for yet new constan ts η > 0 , M > 0, b oth indep endent of s . Therefore, from the deﬁnition of J R and using again Prop osition 4.7 , we obtain ∥ J R − I ∥ L 1 ∩ L ∞ (Γ R \ D τ ν ) ≤ M e − s − η τ ν ∥ ζ 1 / 2 e − η | ζ | ν ∥ L 1 ∩ L ∞ (Γ R \ D τ ν ) = O (e − s − η τ ν ) . (4.34) Estimates ( 4.30 ), ( 4.32 ) and ( 4.34 ) com bined conclude the proof. □ CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 37 R emark 4.19 . F or later reference, observ e that ( 4.31 ) and ( 4.33 ) ma y be combined in to the simpler estimate e − h τ ( ζ | s ) − e − h ∞ ( ζ | s ) = O e − η | ζ | α − s τ κ ! , τ → ∞ , for some α > 0 and η > 0, and v alid uniformly for ζ ∈ R . ▷ W e conclude the asymptotic analysis with the next statemen t. Prop osition 4.20. Fix s 0 ∈ R . F or any κ ∈ (0 , 2) the estimates ∥ R − I ∥ L ∞ ( C \ Γ R ) = O (e − s τ − κ ) , ∥ R ± − I ∥ L ∞ (Γ R ) = O (e − s τ − κ ) , ∥ R ± − I ∥ L 1 ∩ L 2 (Γ R ) = O (e − s τ − κ ) ar e valid as τ → ∞ , uniformly for s ≥ s 0 . Pr o of. Once again, the result follows from standard p erturbation theory of RHPs and the L 1 ∩ L ∞ estimates pro vided by Lemma 4.18 . □ R emark 4.21 . F or admissible Ψ = Ψ τ , the asymptotic expansion in RHP 3.1 –(iii) is v alid as ζ → ∞ in some neigh borho o d of ∞ which in principle ma y dep end on τ . Nev ertheless, thanks to Prop osition 4.20 , Ψ τ ma y b e well-modeled at ζ by Ψ ∞ , and consequently we may choose a uniform (in τ ) neighborho o d of ζ = ∞ on whic h RHP 3.1 –(iii) is v alid for ev ery τ suﬃcien tly large. ▷ Prop osition 4.20 ﬁnishes the asymptotic analysis of Ψ τ , and in the next section we dra w the main consequences that will be useful for later. 5. Consequences of the asymptotic anal ysis for the model problem W e now summarize the consequences of the asymptotic analysis Ψ τ → Ψ ∞ pro vided by Prop osi- tion 4.20 , and results surrounding it. 5.1. Con v ergence of the kernel. Using Cauc h y’s integral formula, we write for any ξ , ζ ∈ C , R ( ξ ) − 1 R ( ζ ) = I + R ( ξ ) − 1 ( R ( ζ ) − R ( ξ )) = I + R ( ξ ) − 1 ζ − ξ 2 π i Z γ R ( u ) ( u − ζ )( u − ξ ) d u, where γ is an y contour encircling b oth ζ and ξ in the coun terclo c kwise direction. Applying Prop osi- tion 4.20 , we obtain R ( ξ ) − 1 R ( ζ ) = I + O (e − s τ − κ ( ζ − ξ )) , τ → ∞ , uniformly in ζ , ξ and s ≥ s 0 , for any κ ∈ (0 , 2). W e unfold ( 4.29 ), concluding that Ψ τ ( ξ ) − 1 Ψ τ ( ζ ) = Ψ ∞ ( ξ ) − 1 ( I + O (e − s τ − κ ( ξ − ζ ))) Ψ ∞ ( ζ ) , τ → ∞ , also uniformly in ζ , ξ and s ≥ s 0 , for an y κ ∈ (0 , 2). Since Ψ ∞ is bounded on compact subsets of C \ { 0 } , and using ( 4.28 ), w e conclude that 1 ξ − ζ  I + e − π i α σ τ ( ξ ) E 21  Ψ τ ( ξ ) − 1 Ψ τ ( ζ )  I − e − π i α σ τ ( ζ ) E 21  21 , + = 1 ξ − ζ  I + e − π i α σ ∞ ( ξ ) E 21  Ψ ∞ ( ξ ) − 1 Ψ ∞ ( ζ )  I − e − π i α σ ∞ ( ζ ) E 21  21 , − + O (e − s τ − κ ) , τ → ∞ , 38 L. MOLAG, G. SIL V A, AND L. ZHANG v alid uniformly in ζ , ξ in compacts of ( −∞ , 0) and s ≥ s 0 , for any κ ∈ (0 , 2), where recall that σ τ / ∞ ( ξ ) = (1 + e − h τ / ∞ ( ζ ) ) − 1 . A comparison with ( 3.47 ) using ( 3.20 ) rev eals the next result. Theorem 5.1. F or any κ ∈ (0 , 2) , the estimate 1 ζ − ξ  I + e − π i α σ τ ( ξ ) E 21  Ψ τ ( ξ ) − 1 Ψ τ ( ζ )  I − e − π i α σ τ ( ξ ) E 21  21 , − = 2 π i e − π i α K α ( − ζ , − ξ ) p σ ∞ ( ξ ) p σ ∞ ( ζ ) + O (e − s τ − κ ) is valid as τ → ∞ , uniformly for ζ , ξ in c omp acts of ( −∞ , 0) and uniformly for s ≥ − s 0 with any ﬁxe d s 0 ∈ R . 5.2. Asymptotics for a relev an t integral. Fix ζ 0 > 0 and in troduce I τ ( s ) . . = Z ∞ s Z 0 − τ 2 ζ 0 e − h τ ( ζ | u ) (1 + e − h τ ( ζ | u ) ) 2  ∆ ζ  Ψ τ ( ζ )  I − e − π i α σ τ ( ζ ) E 21  21 , − d ζ d u, (5.1) where w e recall that ∆ ζ is as explained in ( 2.21 ). This double in tegral will give the leading con tribution to the multiplicativ e statistics ( 2.5 ). If w e apply Theorem 5.1 with ξ → ζ (whic h is p ossible due to the uniformity of the error term therein) and ( 3.48 ), w e obtain as a consequence  ∆ ζ  Ψ τ ( ζ | s )  I − e − π i α σ τ ( ζ | s ) E 21  21 , − = 2 π i e − π i α σ ∞ ( ζ ) K α ( − ζ , − ζ | s ) + o (1) , τ → ∞ , for ζ in ( −∞ , 0) p oin t wise. But w e w an t to ha v e a more precise quan titativ e error estimate, v alid also for ζ near 0 to b e able to perform the integration in ( 5.1 ) when τ → ∞ , and for that we split in to tw o cases, namely ζ < − ε and − ε < ζ < 0. F or ζ < − ε , a com bination of ( 4.29 ) with Prop osition 4.20 yields Ψ τ , − ( ζ | s ) =  I + O  e − s τ κ  Ψ ∞ , − ( ζ | s ) , τ → ∞ , uniformly for ζ < − ε and s ≥ s 0 with an y s 0 ∈ R , and where κ ∈ (0 , 2) is arbitrary . Using this estimate in com bination with Remark 4.19 , w e obtain Ψ τ , − ( ζ | s )  I − e − π i α (1 + e − h τ ( ζ | s ) ) E 21  =  I + O  e − s τ κ  Ψ ∞ , − ( ζ | s ) I − e − π i α 1 + e − h ∞ ( ζ | s ) + O e − s − η | ζ | α τ κ !! E 21 ! , τ → ∞ , with uniform error term as b efore. This iden tit y may b e diﬀeren tiated term by term, and after a cum b ersome but straightforw ard calculation it implies ∆ ζ h Ψ τ , − ( ζ | s )  I − e − π i α (1 + e − h τ ( ζ | s ) ) E 21 i = ∆ ζ h Ψ ∞ , − ( ζ | s )  I − e − π i α (1 + e − h ∞ ( ζ | s ) ) E 21 i + O  e − s τ κ  E 21 [ I + ∆ ζ Ψ ∞ , − ( ζ ) ( I + O (1) E 21 )] . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 39 F rom ( 3.5 ) w e get the rough estimate Ψ ∞ , − ( ζ ) = O ( ζ 1 / 4 − ), v alid for ζ ≤ − ε . T aking the (2 , 1)-entry and using ( 3.48 ) w e up date the abov e to ∆ ζ h Ψ τ ( ζ | s )  I − e − π i α (1 + e − h τ ( ζ | s ) ) E 21 i 21 , − = 2 π i e − π i α σ ∞ ( ζ ) K α ( − ζ , − ζ | s ) + O e − s ζ 1 / 4 − τ κ ! , τ → ∞ , (5.2) v alid, as b efore, uniformly for ζ ≤ − ε , s ≥ s 0 and for any κ ∈ (0 , 2). Next, w e consider − ε ≤ ζ < 0. In this case, using ( 4.24 ), ( 4.29 ) and Proposition 4.20 , we obtain Ψ τ , − ( ζ )  I − e − π i α (1 + e − h τ ( ζ | s ) ) E 21  =  I + O  e − s τ κ  E τ ( ζ ) ( I + λ τ , − ( ζ ) E 21 ) ζ α σ 3 / 2 − ( I + a − ( ζ ) E 12 ) , τ → ∞ , (5.3) v alid uniformly for − ε ≤ ζ < 0 and s ≥ s 0 , and for any κ ∈ (0 , 2). F or the record, w e recall that E τ is as in ( 4.25 ), λ τ is as in ( 4.20 ), and a is as in ( 3.7 ). Using the explicit expression for E τ in ( 4.25 ) and Proposition 3.2 for Ψ = Ψ ∞ , we obtain the represen tation E τ ( ζ ) ( I + λ τ , − ( ζ ) E 12 ) ζ − α σ 3 / 2 ( I + a − ( ζ ) E 12 ) = Ψ ∞ , − ( ζ )  I − e − π i α (1 + e − h ∞ ( ζ | s ) ) E 21  ( I + ( ∗ ) E 12 ) , whic h is v alid algebraically (not asymptotically!) for an y ζ < 0, and where ( ∗ ) represen ts a scalar that do es not aﬀect what comes next. Returning this identit y in ( 5.3 ), we obtain Ψ τ , − ( ζ )  I − e − π i α (1 + e − h τ ( ζ | s ) ) E 21  =  I + O  e − s τ κ  × Ψ ∞ , − ( ζ )  I − e − π i α (1 + e − h ∞ ( ζ | s ) ) E 21  ( I + ( ∗ ) E 12 ) , τ → ∞ , v alid uniformly for − ε ≤ ζ < 0 and s ≥ s 0 as before. The matrix term of the form I + ( ∗ ) E 21 on the righ t-most side do es not aﬀect the ﬁnal result of applying ∆ ζ and taking the (2 , 1)-entry in this identit y . Th us, from this iden tit y and again Theorem 5.1 w e obtain h ∆ ζ h Ψ τ ( ζ | s )  I − e − π i α (1 + e − h τ ( ζ | s ) ) E 21 ii 21 , − = 2 π i e − π i α σ ∞ ( ζ ) K α ( − ζ , − ζ | s ) +  I + e − π i α σ ∞ ( ζ ) E 21  Ψ ∞ ( ζ ) − 1 O  e − s τ κ  Ψ ∞ ( ζ )  I − e − π i α σ ∞ ( ζ ) E 21  21 , − , τ → ∞ . Again thanks to Proposition 3.2 , w e see that Ψ ∞ ( ζ )  I − e − π i α σ ∞ ( ζ ) E 21  = Ψ ∞ , 0 ( ζ ) σ ∞ ( ζ ) − σ 3 / 2 ζ α σ 3 / 2 ( I + a ( ζ ) σ ∞ ( ζ ) E 12 ) = O (1) ζ α σ 3 / 2 ( I + a ( ζ ) σ ∞ ( ζ ) E 12 ) . Using this estimate, the previous estimate simpliﬁes to h ∆ ζ h Ψ τ ( ζ | s )  I − e − π i α (1 + e − h τ ( ζ | s ) ) E 21 ii 21 , − = 2 π i e − π i α σ ∞ ( ζ ) K α ( − ζ , − ζ | s ) + O  e − s ζ α τ κ  , (5.4) 40 L. MOLAG, G. SIL V A, AND L. ZHANG v alid as τ → ∞ , uniformly for − ε ≤ ζ < 0, and uniformly for s ≥ s 0 with any s 0 ∈ R , and where κ ∈ (0 , 2) is arbitrary . Finally , we no w insert ( 5.2 ) and ( 5.4 ) in ( 5.1 ) and use also the uniform deca y of h τ as ζ → −∞ whic h is ensured by Deﬁnition 4.1 –(iii), the conv ergence from Remark 4.21 , and arrive at the following result. Theorem 5.2. F or any κ ∈ (0 , 2) and any s 0 ∈ R , the estimate I τ ( s ) = 2 π i e − π i α Z ∞ s Z 0 −∞ K α ( − ζ , − ζ | u ) e − h ∞ ( ζ | u ) 1 + e − h ∞ ( ζ | u ) d ζ d u + O  e − s τ κ  , τ → ∞ , holds uniformly for s ≥ s 0 . This result ﬁnishes the asymptotic analysis necessary on the mo del problem. 5.3. Pro of of main theorems on integrable equations. In this section w e complete the proof of our main results that connect the limiting probabilistic quan tities to in tegrable equations. They are essentially a recollection of sev eral calculations that we already performed. Pr o of of The or em 2.3 . Deﬁne Φ as in ( 3.32 ). The nonlo cal equation ( 2.11 ) for it w as already obtained in Theorem 3.12 . The asymptotic behavior of Φ as ζ → ±∞ are giv en in ( 3.35 ) and ( 3.36 ). The asymptotic behavior as x → 0 + , in turn, w as obtained in Proposition 4.14 . □ Pr o of of The or em 2.7 . Deﬁne p as in ( 3.26 ). Then, identit y ( 3.45 ) shows that this deﬁnition coincides with the one giv en in ( 2.17 ), and Prop osition 4.14 yields its asymptotic behavior as x → 0 + as claimed b y Theorem 2.7 . It remains to show the connection with L (Bes) α . F or that, w e start from ( 3.48 ) and use ( 3.21 ) and ( 3.29 ) to express K α ( − ζ , − ζ | s , x ) = e π i α u − 1 /m σ ∞ ( ζ ) 2 π i h Υ ( u 1 /m ζ ) − 1 Υ ′ ( u 1 /m ζ ) i 21 , − . P erforming the change of v ariables ξ = − u 1 /m ζ = − 4 x 2 ζ in ( 2.16 ), we obtain from ( 3.20 ) that − log L (Bes) α ( s , x ) = e π i α 2 π i Z ∞ s Z 0 −∞  Υ ( ξ | s = u, x ) − 1 Υ ′ ( ξ | s = u, x )  21 , − ∂ s σ Φ ( ξ | s = u )d ξ d u. (5.5) Observ e that σ Φ is indep enden t of x , recall ( 2.10 ). A direct calculation from the jump of Υ (which coincides with the jump of Φ ∞ in ( 3.22 )) sho ws that [ Υ − 1 Υ ′ ] 21 , − = e − 2 π i α [ Υ − 1 Υ ′ ] 21 , + . Using then ( 3.31 ), ( 3.32 ) and ( 3.34 ), we get  Υ − 1 Υ ′  21 , − = i e − 2 π i α  Φ ∂ x Φ ′ − Φ ′ ∂ x Φ  . T aking one x -deriv ative and using ( 3.33 ), we get that the x -deriv ativ e of the righ t-hand side of the ab o v e form ula is giv en b y i e − 2 π i α Φ 2 . Using this relation in ( 5.5 ), we see that − ∂ x  log L (Bes) α ( s , x )  = e − π i α 2 π Z ∞ s Z 0 −∞ Φ( ξ | s = u, x ) 2 ∂ s σ Φ ( ξ | s = u )d ξ d u. When obtaining this identit y , w e exchanged the deriv ativ es with the integrals, a calculation which is justiﬁed b y the bounds obtained in ( 4.8 )–(ii) and the exp onen tial deca y of σ Φ . F rom the deﬁnition of p in ( 2.17 ) it follows that this identit y is equiv alen t to ( 2.19 ), concluding the pro of of Theorem 2.7 . □ CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 41 Pr o of of The or em 2.8 . Supp ose now that m = 1. In particular, from ( 2.10 ) w e see that σ Φ ( ζ ) = 1 / (1 + e − s + ζ ). By setting b Φ ∞ ( ζ ) . . =  1 0 − i sx 2 1  Φ ∞ ( ζ + s ) , it then follows from RHP 3.11 , ( 3.26 ) and direct calculations that b Φ ∞ satisﬁes the following RHP . RHP 5.3. Find a 2 × 2 matrix-v alued function b Φ ∞ : C \ ( −∞ , − s ] → C 2 × 2 with the following prop erties. (i) b Φ ∞ is analytic on C \ ( −∞ , − s ]. (ii) The en tries of b Φ ∞ , ± are con tin uous along ( −∞ , − s ) and satisfy b Φ ∞ , + ( ζ ) = b Φ ∞ , − ( ζ )  e π i α 1 1+e ζ 0 e − π i α  , ζ < 0 . (5.6) (iii) As ζ → ∞ , we ha ve b Φ ∞ ( ζ ) = I + b Φ ∞ , 1 ζ + O ( ζ − 2 ) ! ζ − σ 3 / 4 U 0 e x ζ 1 / 2 σ 3  I + (e π i α χ S + ( ζ ) − e − π i α χ S − ( ζ )) E 21  , where b Φ ∞ , 1 =  1 0 − i sx 2 1   q − s 4 − i p i r − q + s 4   1 0 i sx 2 1  + s 2 x 2 8 − i sx 2 − i 24 s 2 x ( sx 2 + 3) − s 2 x 2 8 ! . (iv) As ζ → − s , we ha ve b Φ ∞ ( ζ ) = b Φ ∞ , 0 ( ζ )( ζ + s ) α σ 3 / 2 ( I + a ( ζ + s ) E 12 ) (1 + e ζ ) σ 3 / 2 , where b Φ ∞ , 0 ( ζ ) =  1 0 − i sx 2 1  Φ ∞ , 0 ( ζ + s ) (5.7) is analytic near ζ = − s and a is as in ( 3.7 ). Note that ( b Φ ∞ , 1 ) 12 = − i( p + sx 2 ), w e deﬁne, similar to ( 3.29 ), b Υ ( ζ ) . . =  I + i  p + sx 2  E 21  b Φ ∞ ( ζ ) . Th us, b Υ satisﬁes the ODE ∂ x b Υ ( ζ ) = b B ( ζ ) b Υ ( ζ ) , where b B ( ζ ) =  0 − i i( ζ + 2 ∂ x p + s ) 0  . (5.8) Since the jump matrix of b Φ ∞ in ( 5.6 ) is independent of s as well, we obtain from ( 5.7 ) that ∂ s b Υ ( ζ ) = b A ( ζ ) b Υ ( ζ ) , where b A ( ζ ) =  0 0 i( ∂ s p + x 2 ) 0  + 1 ζ + s  a b c − a  (5.9) for some functions a , b and c dep ending on x and s . The compatibility condition ∂ 2 sx b Υ = ∂ 2 xs b Υ yields the zero curv ature equation ∂ s b B − ∂ x b A = b A b B − b B b A . 42 L. MOLAG, G. SIL V A, AND L. ZHANG Inserting ( 5.8 ) and ( 5.9 ) in to the ab ov e equation and comparing the coeﬃcients of O (1) and O (1 / ( ζ + s )) terms, w e arrive at b = − i  ∂ s p + x 2  , a = − 1 2  ∂ 2 sx p + 1 2  , c = i ∂ x a − 2 b ∂ x p , (5.10) and ∂ x c = 4i a ∂ x p . (5.11) F rom ( 5.10 ), it is readily seen that c = i  − 1 2 ∂ 3 sxx p + 2 ∂ x p ∂ s p + x ∂ x p  . This, together with ( 5.11 ), implies the PDE for p claimed in Theorem 2.8 . □ 6. Asymptotic anal ysis of the RHP for OPs Our eﬀort so far was the in-depth study of the mo del problem introduced in Section 3.1 . No w w e come back to our original mo del, that is, extracting large n information on the p oint pro cess with joint distribution ( 2.1 ). In this section, w e carry out the asymptotic analysis of the asso ciated polynomials, and in this analysis the solution Ψ τ of the mo del problem with admissible data, as well as its limit Ψ ∞ , will play a cen tral role. 6.1. Equilibrium measures and related quan tities. In the course of the asymptotic analysis, we will need certain properties of equilibrium measures that w e now brieﬂy review. The prop erties we now mention can b e found, e.g., in [ 31 , 47 ], see also [ 2 ] for the speciﬁc structure of Cauc hy transforms of log equilibrium measures with hard edge. The e quilibrium me asur e of the interv al [0 , + ∞ ) in the external ﬁeld V is the unique probabilit y measure supported on [0 , + ∞ ) that minimizes the functional µ 7→ Z Z log 1 | x − y | d µ ( x )d µ ( y ) + Z V ( x )d µ ( x ) o v er all Borel probability measures µ supp orted on [0 , + ∞ ). This measure is uniquely characterized by the Euler-L agr ange variational c onditions: there exists a constan t ℓ V ∈ R for whic h Z log 1 | x − y | d µ V ( y ) + 1 2 V ( x ) + ℓ V ( = 0 , x ∈ supp µ V , ≥ 0 , x ∈ [0 , + ∞ ) . (6.1) Under our assumptions, the equilibrium measure µ V is absolutely contin uous with respect to the Leb esgue measure, say d µ V ( x ) = µ ′ V ( x )d x , for some function µ ′ V . The regularit y conditions on µ V that w ere men tioned in Assumption 2.1 translate in to the follo wing precise conditions. Assumption 6.1 (One-cut regularit y condition) . W e assume that the p oten tial V is suc h that the follo wing holds. (i) F or some a > 0, supp µ V = [0 , a ]. (ii) The inequalit y in ( 6.1 ) is strict, that is, Z log 1 | x − y | d µ V ( y ) + 1 2 V ( x ) + ℓ V > 0 , x ∈ ( a, ∞ ) . (6.2) (iii) µ ′ V ( x ) > 0 for x ∈ (0 , a ). CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 43 (iv) F or some constants κ 0 , κ a > 0, w e ha v e µ ′ V ( x ) = κ 0 √ x (1 + O ( x )) , x ↘ 0 , and µ ′ V ( x ) = κ a √ a − x (1 + O ( x − a )) , x ↗ a. F or the next transformation, we need to consider quantities related to the equilibrium measure. F or C µ V ( z ) . . = Z d µ V ( s ) s − z , z ∈ C \ [0 , a ] , b eing the Cauch y transform of the equilibrium measure, w e associate to it its ϕ function, ϕ ( z ) . . = Z z 0  C µ V ( s ) + 1 2 V ′ ( s )  d s, z ∈ C \ ( −∞ , a ] , (6.3) with the path of in tegration starting on the upp er half-plane. Under Assumption 6.1 , the Cauc h y transform satisﬁes an algebraic equation of the form  C µ V ( z ) + V ′ ( z ) 2  2 = z − a z h V ( z ) 2 , for some p olynomial h V with h V (0) , h V ( a )  = 0. Standard argumen ts show that ϕ + ( z ) + ϕ − ( z ) = 0 , 0 < z < a, ϕ + ( z ) − ϕ − ( z ) = 2 π i µ V ([0 , z ]) ∈ i R + , 0 < z < a. (6.4) In particular, h V ( x ) < 0 on (0 , a ), and µ ′ V ( x ) = 1 π r a − x x | h V ( x ) | , 0 < x < a. F urthermore, a combination of ( 6.4 ), Cauc h y-Riemann equations and the inequality ( 6.2 ) yields that ϕ ( x ) > 0 for x > a, and Re ϕ ( z ) < 0 for z slightly ab ov e or below (0 , a ) . (6.5) A local analysis based on conformal prop erties of ϕ sho ws that for some δ > 0, Re ϕ ( z ) < 0 for z ∈ D δ \ [0 , δ ) and ϕ ( z ) < 0 for − δ < z < 0 . Consequen tly , the function ψ ( z ) . . = 1 4 ϕ ( z ) 2 is a conformal map in a neighborho o d of the origin, and the b ehavior near the origin in Assumption 6.1 – (iv) ensures that ψ ( z ) = − c V z (1 + O ( z )) , z → 0 , for c V . . = π 2 κ 2 0 , (6.6) and where w e recall that κ 0 > 0 is as in Assumption 6.1 –(iv). This deﬁnition of c V is consisten t with ( 2.14 ). Finally , a direct calculation sho ws that, as z → ∞ , ϕ ( z ) = V ( z ) 2 + ℓ V − log z + ϕ ∞ z + O ( z − 2 ) , (6.7) for some ϕ ∞ ∈ R , where ℓ V is as in Assumption 6.1 –(ii). 44 L. MOLAG, G. SIL V A, AND L. ZHANG 6.2. The RHP for OPs. W e start with the RHP for the corresp onding OPs. RHP 6.2. Find a 2 × 2 matrix-v alued function Y : C \ [0 , + ∞ ) → C 2 × 2 with the following prop erties. (i) Y : C \ [0 , ∞ ) → C 2 × 2 is analytic. (ii) The matrix Y has contin uous b oundary v alues Y ± along (0 , + ∞ ), and they are related by Y + ( x ) = Y − ( x ) J Y ( x ), x > 0, with J Y ( x ) . . = I + ω n ( x ) E 12 , x > 0 . (iii) As z → ∞ , Y ( z ) =  I + 1 z Y 1 + O ( z − 2 )  z n σ 3 . (iv) As z → 0, Y ( z ) = O  1 h α ( z ) 1 h α ( z )  , where the O -term is to b e read entry-wise and here and from now on w e set h α ( z ) . . =      z α , α < 0 , log z , α = 0 , 1 , α > 0 . (6.8) The weigh t ω n ( x ) = ω n ( x | s ) w as in troduced explicitly in ( 2.2 ) and dep ends on s , and consequen tly so do es Y ( z ) = Y ( z | s ) and an y other quantit y associated to the orthogonal p olynomials, but sometimes w e omit this dep endence from our notation. As it is no w classical (see [ 34 ]), the solution to this RHP is related to the monic OP P j of degree j for the weigh t ω n via Y 11 ( z ) = P n ( z ) and Y 21 ( z ) = − 2 π i( γ ( n ) n − 1 ) 2 P n − 1 ( z ) . As a consequence of the Christoﬀel-Darb oux form ula, K n ( x, y ) = 1 2 π i( x − y ) e T 2 Y + ( y ) − 1 Y + ( x ) e 1 , x  = y , and K n ( x, x ) = 1 2 π i e T 2 Y + ( x ) − 1 Y ′ + ( x ) e 1 , (6.9) where e 1 . . = (1 , 0) T and e 2 . . = (0 , 1) T are the canonical base vectors for R 2 . Using the explicit form ∂ u ω n ( x | u ) = e − u − n 2 m Q ( x ) σ n ( x | u ) ω n ( x | u ) = ω n ( x | u ) ∂ u log σ n ( x | u ) , the iden tit y given by ( 2.9 ) updates to log L Q n ( s ) = − 1 2 π i Z ∞ s Z ∞ 0  Y ( x | u ) − 1 Y ′ ( x | u )  21 , + x α e − nV ( x ) ∂ u σ n ( x | u )d x d u = − 1 2 π i Z ∞ s Z ∞ 0  Y ( x | u ) − 1 Y ′ ( x | u )  21 , + ω n ( x | u ) ∂ u log σ n ( x | u )d x d u. (6.10) This deformation formula will b e the starting p oint for our asymptotic analysis of L Q n ( s ), but ﬁrst we need to obtain asymptotics for Y . T o obtain asymptotics for Y we once again emplo y the Deift-Zhou nonlinear steep est descent metho d. The transformations needed in this case are no w canonical, so we go o ver them brieﬂy . When compared with more standard situations the only ma jor obstacle is the construction of a lo cal parametrix near the origin, whic h appears in virtue of the factor σ n whic h scales singularly near the origin. In this construction, w e will use the mo del problem that w e previously studied in Sections 3 – 5 . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 45 0 a L + L − Figure 2. The con tour Γ S for RHP 6.4 . 6.3. First transformation: introduction of the g -function. The ﬁrst transformation consists in in troducing the g -function, which here is replaced b y the ϕ - function from ( 6.3 ). T ransform T ( z ) . . = e − nℓ V σ 3 Y ( z ) e n ( ϕ ( z ) − V ( z ) / 2) σ 3 , z ∈ C \ [0 , ∞ ) . Then T satisﬁes the following RHP . RHP 6.3. Find a 2 × 2 matrix-v alued function Y : C \ [0 , + ∞ ) → C 2 × 2 with the following prop erties. (i) T : C \ [0 , ∞ ) → C 2 × 2 is analytic. (ii) The matrix T has con tin uous b oundary v alues T ± along (0 , + ∞ ), and they are related by T + ( x ) = T − ( x ) J T ( x ), x > 0, with J T ( x ) . . =  e n ( ϕ + ( x ) − ϕ − ( x )) σ n ( x ) x α e − n ( ϕ + ( x )+ ϕ − ( x )) 0 e − n ( ϕ + ( x ) − ϕ − ( x ))  , x > 0 . (iii) As z → ∞ , T ( z ) = I + 1 z T 1 + O ( z − 2 ) . with T 1 . . = e − nℓ V σ 3 Y 1 e nℓ V σ 3 + nϕ ∞ σ 3 . (iv) As z → 0, T ( z ) = O  1 h α ( z ) 1 h α ( z )  , where w e recall that h α is giv en in ( 6.8 ). 6.4. Second transformation: op ening of lenses. The next step is to op en lenses L ± around the interv al [0 , a ]. F or that, we mo dify S ( z ) . . =      T ( z ) I ∓ z − α e 2 nϕ ( z ) σ n ( z ) E 21 ! , z ∈ L ± , T ( z ) , otherwise . 46 L. MOLAG, G. SIL V A, AND L. ZHANG In the ab o v e and what follo ws, z α is alw ays the principal branch, with branc h cut on ( −∞ , 0]. In tro duce the new contour Γ S . . = [0 , ∞ ) ∪ ∂ L + ∪ ∂ L − . Using the prop erties of the function ϕ discussed in Section 6.1 , in particular the jump relations ( 6.4 ), it follo ws that the matrix S solves the follo wing RHP . RHP 6.4. Find a 2 × 2 matrix-v alued function S : C \ Γ S → C 2 × 2 with the following prop erties. (i) S : C \ Γ S → C 2 × 2 is analytic. (ii) The matrix S has contin uous b oundary v alues S ± along the open subarcs of Γ S , and they are related b y S + ( z ) = S − ( z ) J S ( z ), z ∈ Γ S \ { 0 , a } , with J S ( z ) . . =              z α σ n ( z ) E 12 − z − α σ n ( z ) E 21 , 0 < z < a, I + z − α σ n ( z ) e 2 nϕ ( z ) E 21 , z ∈ ∂ L ± \ R , I + σ n ( z ) z α e − 2 nϕ ( z ) E 12 , z > a. (iii) As z → ∞ , S ( z ) = I + 1 z S 1 + O ( z − 2 ) . with S 1 . . = T 1 . (iv) As z → 0, the matrix S has the behavior S ( z ) =            O 1 h α ( z ) 1 h α ( z ) ! , outside the lens, O h − α ( z ) h α ( z ) h − α ( z ) h α ( z ) ! , inside the lens, with h α giv en in ( 6.8 ). Thanks to ( 6.5 ), w e ha ve that J S ( z ) → I for z ∈ Γ S \ [0 , a ] in an appropriate sense, and we now pro ceed to the construction of global and lo cal parametrices. These parametrices are matrix-v alued functions that satisfy the jumps for S in the interv al (0 , a ) and in neighborho o ds of z = 0 , a , and will pro duce goo d appro ximations for S near these sets. 6.5. The global parametrix. The global parametrix RHP is obtained after neglecting the exp onentially small jumps of S . Con- cretely , the global parametrix G is the solution to the follo wing RHP . RHP 6.5. Find a 2 × 2 matrix-v alued function G : C \ [0 , a ] → C 2 × 2 with the following prop erties. (i) G : C \ [0 , a ] → C 2 × 2 is analytic. (ii) The matrix G has con tin uous b oundary v alues G ± along [0 , a ], and they are related by G + ( z ) = G − ( z )  z α σ n ( z ) E 12 − z − α σ n ( z ) E 21  , 0 < z < a. (iii) As z → ∞ , G ( z ) = I + O ( z − 1 ) . (iv) The matrix G has at worse L 2 -in tegrable singularities at z = 0 , a . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 47 The construction of the solution G is standard, and we now describe it. As w e will see, G dep ends on σ n , so in fact G = G n , and we will also need to understand the b ehavior of G n as n → ∞ , whic h w e also describ e in this section. The solution G is explicitly giv en by G ( z ) . . = e − p 0 σ 3 D σ 3 ∞ M ( z ) D ( z ) − σ 3 e p ( z ) σ 3 , z ∈ C \ [0 , a ] , (6.11) where eac h of the quantities is as follo ws. The function D is analytic on C \ [0 , a ], and satisﬁes D + ( z ) D − ( z ) = z α for z ∈ (0 , a ) , and D ( z ) → D ∞  = 0 as z → ∞ . With standard metho ds, w e ﬁnd D ( z ) . . = z α/ 2 d ( z ) α/ 2 , z ∈ C \ [0 , a ] , where d ( z ) is the conformal map from C \ [0 , a ] to the exterior of the unit disk, normalized to satisfy lim x → + ∞ x − 1 d ( x ) = . . d ∞ > 0. A simple calculation sho ws that d ( z ) = 2 a z − 1 + 2 a z 1 / 2 ( z − a ) 1 / 2 , z ∈ C \ [0 , a ] , with principal branch of the ro ot, so that d ( z ) = 4 z /a + O (1) as z → ∞ and therefore d ∞ = 4 /a . Th us, D ∞ = 2 − α a α/ 2 . The function p ( z ) solves the scalar RHP p + ( z ) + p − ( z ) = − log σ n ( z ) , 0 < z < a, with behavior p ( z ) = O (1) , z → 0 , a, ∞ . With standard metho ds, w e ﬁnd p ( z ) . . = ( z ( z − a )) 1 / 2 2 π Z a 0 log( σ n ( x )) p | x ( x − a ) | d x x − z , z ∈ C \ [0 , a ] , with principal branch of the square ro ot. The constan t p 0 is determined from p ( z ) = p 0 + p 1 z + O ( z − 2 ) , z → ∞ , from whic h we ﬁnd p 0 . . = − 1 2 π Z a 0 log( σ n ( x )) p | x ( x − a ) | d x, p 1 . . = − 1 2 π Z a 0 x log( σ n ( x )) p | x ( x − a ) | d x + a 2 p 0 . F or U 0 . . = 1 √ 2  1 i i 1  , g ( z ) . . = z − a z , the matrix M is M ( z ) = U 0 g ( z ) σ 3 / 4 U − 1 0 , (6.12) whic h solv es the jump M + ( z ) = M − ( z )( E 12 − E 21 ) . The function p = p ( · | n ) and the constan t p 0 = p 0 ( n ) b oth v ary with n , but we mostly omit this dep endence in our notation. It is how ever important to establish its b ehavior as n → ∞ . Using Prop osition B.1 we estimate these terms, the result is summarized in the next prop osition. 48 L. MOLAG, G. SIL V A, AND L. ZHANG Prop osition 6.6. Set p ∞ ( s ) . . = a − 1 / 2 t − 1 / (2 m ) 2 π m F 1 2 m − 1 ( s ) , wher e F β ( s ) = − β Γ( β ) Li β +2 ( − e − s ) with Li β b eing the p olylo garithms. F or any s 0 ∈ R ﬁxe d, the estimate p 0 = 1 n p ∞ ( s ) + O ( n − 1 − 2 m ) , n → ∞ , holds uniformly for s ≥ s 0 , and the estimate p ( z ) =  z − a z  1 / 2 1 n p ∞ ( s ) + O ( n − 1 − 2 m ) , n → ∞ holds uniformly for s ≥ s 0 , and also uniformly for z in c omp acts of C \ { 0 , a } , wher e for z ∈ (0 , a ) it should b e understo o d for b oundary values p ± ( z ) , p ∞ , ± . Pr o of. W e may estimate p 0 and p using Prop osition B.1 –(ii), once we iden tify β = − 1 2 , Q ( x ) = tq ( x ), and resp ectively f ( x ) = | x − a | − 1 / 2 and f ( x ) = | x − a | − 1 / 2 ( x − z ) − 1 (or rather w e lo ok at the real and imaginary part of f ). In the case that z ∈ (0 , a ), the reader may verify that the (half ) residue con tribution corresp onding to x = z to the in tegral deﬁning p ( z ) does not con tribute to the dominan t order, and one ma y eﬀectiv ely apply Proposition B.1 –(ii) for some δ < z . W e obtain the stated result after in v oking Prop osition B.1 –(i). □ As a consequence, w e hav e the following prop osition. Prop osition 6.7. The glob al p ar ametrix G deﬁne d in ( 6.11 ) satisﬁes G ( z ) =  I + O ( n − 1 − 2 m )  D σ 3 ∞ M ( z ) D ( z ) − σ 3 , n → ∞ , uniformly for z in c omp acts of C \ { a, 0 } , wher e for z ∈ (0 , a ) this r elation should b e understo o d as valid (uniformly) for either of the ± -b oundary values. In p articular, the b oundary values G ± r emain uniformly b ounde d on c omp acts of (0 , a ) . Pr o of. Prop osition 6.6 ensures that as n → ∞ , e − p 0 σ 3 = I + O ( n − 1 − 2 m ) , as w ell as e p ( z ) σ 3 = I + O ( n − 1 − 2 m ) , with the latter b eing v alid uniformly for z ∈ C \ { 0 , a } in compacts. Because D ∞ , D and M are indep enden t of n and are b ounded on compacts of C \ { 0 , a } , we then ha ve that D σ 3 ∞ M ( z ) D ( z ) − σ 3 e p ( z ) σ 3 =  I + O ( n − 1 − 2 m )  D σ 3 ∞ M ( z ) D ( z ) − σ 3 , and the result follo ws from the explicit expression ( 6.11 ). □ With the global parametrix now constructed, w e mo v e forward to the local parametrices. 6.6. The lo cal parametrix near the hard edge. The local parametrix P (0) , constructed in a neighborho o d U 0 of z = 0, reads as follo ws. RHP 6.8. Find a 2 × 2 matrix-v alued function P (0) : U 0 \ Γ S → C 2 × 2 with the following prop erties (i) P (0) is analytic on U 0 \ Γ S . (ii) It has jump P (0) + ( z ) = P (0) − ( z ) J S ( z ), z ∈ Γ S ∩ U 0 . (iii) F or z ∈ ∂ U 0 , it b ehav es as P (0) ( z ) = ( I + o (1)) G ( z ) , n → ∞ . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 49 Making the change L ( z ) = P (0) ( z ) e − nϕ ( z ) σ 3 ( − z ) α σ 3 / 2 , z ∈ U 0 \ Γ S , (6.13) with the principal branc h of the root ( · ) α/ 2 , it follows that L should satisfy the follo wing RHP . RHP 6.9. Find a 2 × 2 matrix-v alued function L : U 0 \ Γ S → C 2 × 2 with the following prop erties. (i) L is analytic on U 0 \ Γ S . (ii) It has jump L + ( z ) = L − ( z ) J L ( z ), z ∈ Γ S ∩ U 0 , with J L ( z ) . . =        σ n ( z ) E 12 − 1 σ n ( z ) E 21 , z > 0 , I + e ∓ π α i σ n ( z ) E 21 , z ∈ ( ∂ L ± ∩ U 0 ) \ [0 , ∞ ) , (iii) F or z ∈ ∂ U 0 , it b ehav es as L ( z ) = ( I + o (1)) G ( z )( − z ) α σ 3 / 2 e − nϕ ( z ) σ 3 , n → ∞ . The next step is to use the conformal map ψ from ( 6.6 ) to translate the RHP in the z -plane to a RHP in the transformed plane. F or that, we lo ok at ζ . . = n 2 ψ ( z ) , that is, z = ψ − 1  ζ n 2  , as a conformal c hange of v ariables. This corresp ondence satisﬁes z < 0 if, and only if, ζ > 0 . This w a y , it follows that 2 ζ 1 / 2 = 2( n 2 ψ ( z )) 1 / 2 = n ( ϕ ( z ) 2 ) 1 / 2 = ± nϕ ( z ) for the principal branc h of the log and some c hoice of the sign. Because ϕ ( z ) < 0 for z < 0, w e hav e in fact 2 ζ 1 / 2 = − nϕ ( z ) . W e also ha ve to account for the transformation of the jump from the z -plane to the ζ -plane. F or that, for ψ − 1 b eing the inv erse of ψ , we take δ > 0 such that ψ − 1 is analytic on the disk D 2 δ (0) in the w plane, and suc h that ψ ( U 0 ) = D δ (0). W e also mak e sure that the lips of the lenses are mapp ed to the con tours Γ ± from ( 3.1 ). In tro duce the new function H Q ( w ) . . = Q ( ψ − 1 ( w )) , | w | < δ, whic h is an analytic function on D δ (0) that satisﬁes H Q ( w ) = ( − 1) m t c m V w m (1 + O ( w )) , w → 0 . In particular, the function h n ( ζ ) . . = s + n 2 m Q ( z ) = s + n 2 m Q  ψ − 1  ζ n 2  = s + n 2 m H Q  ζ n 2  , is analytic on | ζ | ≤ δ n 2 and satisﬁes h n ( ζ ) = s + ( − 1) m u ζ m + O ( n − 2 ) , n → ∞ , u . . = t c m V > 0 , uniformly for ζ in compacts of C , or also the more reﬁned estimate h n ( ζ ) = s + ( − 1) m u ζ m + O ( ζ m +1 n − 2 ) , n → ∞ , 50 L. MOLAG, G. SIL V A, AND L. ZHANG whic h is v alid for | ζ | ≤ δ n 2 . W e extend H Q in a C ∞ w a y to a neighborho o d of Γ, so that with the identiﬁcation τ = n , this function h n is admissible in the sense of Deﬁnition 4.1 . In particular, from the considerations ab ov e and ( 4.28 ), | σ n ( x ) − σ ∞ ( ζ ( x )) | = O (e − s n − 2 ) , n → ∞ , σ ∞ ( ζ ) . . = 1 1 + e − h ∞ ( ζ ) , (6.14) uniformly for ζ in compacts of C , where we recall that h ∞ ( ζ ) = s + ( − 1) m u ζ m is as in ( 3.18 ). Mo ving forw ard with the construction for L , set E n ( z ) . . = E 1 ( z ) n σ 3 / 2 , E 1 ( z ) . . = D σ 3 ∞ M ( z )( − z ) α σ 3 / 2 D ( z ) − σ 3 U 0 ( ψ ( z )) σ 3 / 4 , z ∈ U δ . (6.15) A direct calculation from its deﬁnition sho ws that E 1 is analytic near z = 0. With all these deﬁnitions, we construct the solution L in the form L ( z ) = E n ( z ) b Ψ n ( z ) , b Ψ n ( z ) = σ 3 Ψ n ( ζ ) σ 3 , ζ = ζ ( z ) = n 2 ψ ( z ) , (6.16) where Ψ n should satisfy the follo wing RHP . RHP 6.10. Find a 2 × 2 matrix-v alued function Ψ n : C \ Γ → C 2 × 2 with the following prop erties. (i) Ψ n is analytic on C \ Γ. (ii) The jump matrix J n for Ψ n is J n ( ζ ) . . =          I + (1 + e − h n ( ζ ) ) e π α i E 21 , ζ ∈ Γ 1 , 1 1 + e − h n ( ζ ) E 12 − (1 + e − h n ( ζ ) ) E 21 , ζ ∈ Γ 2 , I + (1 + e − h n ( ζ ) ) e − π α i E 21 , ζ ∈ Γ 3 , (iii) As ζ → ∞ , Ψ n ( ζ ) = ζ − σ 3 / 4 U 0  I + O ( ζ − 1 / 2 )  e 2 ζ 1 / 2 σ 3 , (6.17) where the error term is v alid uniformly for | ζ | > R , for some R > 0 independent of n . A direct comparison sho ws that, with the iden tiﬁcation n = τ , the solution Ψ n of the model RHP 3.1 is a solution Ψ n of this RHP . In fact, RHP 6.10 (i) and (ii) are in direct corresp ondence with RHP 3.1 , and a direct calculation shows that RHP 3.1 –(iii) implies RHP 6.10 –(iii) (see also Remark 4.21 ). F rom no w on, w e refer to this choice Ψ n as b eing the solution to RHP 6.10 , even though RHP 6.10 as posed ma y admit more than one solution 3 . In particular, Prop osition 3.2 ensures that Ψ n ( ζ ) = Ψ ( n ) 0 ( ζ ) ζ α σ 3 / 2 ( I + a ( ζ ) E 12 )  I − ( χ S + ( ζ ) e π i α − χ S − ( ζ ) e − π i α ) E 21  × (1 + e − h n ( ζ ) ) σ 3 / 2 , ζ → 0 , (6.18) where Ψ ( n ) 0 is analytic near the origin. With the found L , we no w reco v er P (0) from ( 6.13 ). W e now collect some prop erties that will pla y some role later on. Prop osition 6.11. The fol lowing pr op erties hold. (i) The matrix E 1 is indep endent of n , and it is analytic in a neighb orho o d of z = 0 . 3 T o ensure unique solution, we would hav e to sp ecify the b ehavior as ζ → 0, but for clarity of presen tation we opt for the approac h presented without this condition a priori, only a p osteriori CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 51 (ii) F or any s 0 ∈ R , the function P (0) satisﬁes P (0) ( z ) =  I + O ( n − 1 )  G ( z ) , n → ∞ , (6.19) uniformly for z ∈ ∂ U δ , and uniformly for s ≥ s 0 . (iii) The entries of the matrix Ψ n, ± ( ζ ) ± 1 r emain b ounde d as n → ∞ , uniformly for ζ ∈ R with 1 / M ≤ | ζ | ≤ M and s ≥ s 0 for any s 0 ∈ R and any M > 0 . (iv) F or any s 0 ∈ R ﬁxe d, ther e exist n 0 > 0 and M > 0 , for which the matrix Ψ ( n ) 0 is analytic and uniformly b ounde d on | ζ | < 1 / M , for any n ≥ n 0 and any s ≥ s 0 . (v) F or any s 0 > 0 ﬁxe d, ther e exist n 0 > 0 and M > 0 for which the exp ansion ( 6.17 ) is valid on | ζ | > M , uniformly for n ≥ n 0 and s ≥ s 0 . Pr o of. F rom its explicit expression, it is immediate that E 1 is indep enden t of n and it is analytic on U δ \ [0 , a ]. A straigh tforward calculation sho ws that, in addition, it has no jumps along (0 , a ), so its singularit y at z = 0 is isolated. F rom the b eha vior of its en tries, w e see that in fact this singularit y is remo v able, and the analyticity of E 1 follo ws. The estimate on L claimed in (ii) follo ws from the corresp ondence ζ = n 2 ψ ( z ), the RHP satisﬁed b y Ψ n , namely RHP 3.1 –(iii), and Prop osition 6.7 . W e skip details. The claims (iii) and (iv) are true if w e replace Ψ n b y Ψ ∞ . Since Ψ n → Ψ ∞ uniformly on compacts, standard argumen ts show that they are true for Ψ n as w ell, as long as w e tak e n suﬃcien tly large. Finally , claim (v) is the same as Remark 4.21 for Ψ τ = Ψ n . □ These considerations ﬁnish the construction of the lo cal parametrix. 6.7. The lo cal parametrix near the soft edge. The lo cal parametrix P ( a ) near the soft edge z = a is constructed in the usual w a y . Its main building blo c k is the RHP for the Airy function that we describ e next. Set Γ A . . = R ∪ (e 2 π i / 3 ∞ , 0] ∪ (e − 2 π i / 3 ∞ , 0] , with the orien tation of the arcs on real axis to b e the natural orientation induced b y R , and with the non-real arcs oriented from ∞ to 0. The Airy parametrix A is the solution to the following RHP . RHP 6.12. Find a 2 × 2 matrix-v alued function A : C \ Γ A → C 2 × 2 with the following prop erties. (i) A is analytic on C \ Γ A . (ii) The jump condition A + ( ζ ) = A − ( ζ ) J A ( ζ ), ζ ∈ Γ A holds, where the jump matrix J A for A is J A ( ζ ) . . =      I + E 12 , ζ ∈ (0 , ∞ ) , I + E 21 , ζ ∈ (e ± 2 π i / 3 , 0) , E 12 − E 21 , ζ ∈ ( −∞ , 0) . (iii) As ζ → ∞ , A ( ζ ) = ζ − σ 3 / 4 U 0  I + O ( ζ − 3 / 2 )  e − 2 3 ζ 3 / 2 σ 3 . (6.20) The solution A can b e constructed explicitly using Airy functions [ 27 ], but w e will not need its explicit expression. W e now explain how to use it in our framework to construct the lo cal parametrix near z = a . 52 L. MOLAG, G. SIL V A, AND L. ZHANG Since we are assuming Q to b e analytic on a neighborho o d of the p ositive real axis, the function σ n admits an analytic contin uation to a neighborho o d of z = a . F urthermore, its p oles are the solutions to Q ( z ) = − s n 2 m + (2 k + 1) π i n 2 m . In particular, assuming that s ≥ − s 0 , with s 0 > 0 ﬁxed, w e see that there are no p oles of σ n near z = a . F urthermore, Q ( a ) > 0, and therefore for the same range of v alues of s , s + n 2 m Re Q ( z ) ≥ η n 2 m , for some η > 0, on any (small enough) neigh b orho o d D δ ( a ) of a . In particular, this means that the functions σ n and σ − 1 n b oth admit an analytic square ro ot on a neigh b orho o d D δ ( a ) indep endent of n and s , and for this square root, σ n ( z ) σ 3 / 2 = I + O (e − η n 2 m ) , n → ∞ , uniformly for z ∈ D δ ( a ) and also uniformly for s ≥ − s 0 . Let us also in tro duce e ϕ ( z ) . . = ϕ ( z ) + π i and e ψ ( z ) . . =  3 2 e ϕ ( z )  2 / 3 . With standard arguments, we see that e ψ is a conformal map from a neigh b orho o d of z = a to a neigh b orho o d of the origin, and it maps ( a, a + δ ) to the p ositive real axis. With this in mind, the lo cal parametrix near z = a takes the form P ( a ) ( z ) . . = E ( a ) n ( z ) A ( n 2 / 3 e ψ ( z )) e n e ϕ ( z ) σ 3 σ n ( z ) − σ 3 / 2 z − α σ 3 / 2 , (6.21) with E ( a ) n ( z ) . . = E ( a ) 1 ( z ) n σ 3 / 6 , E ( a ) 1 ( z ) . . = D σ 3 ∞ M ( z ) D ( z ) − σ 3 z α σ 3 / 2 U − 1 0 e ψ ( z ) σ 3 / 4 . (6.22) Also, it is straightforw ard to show that this factor E ( a ) 1 is analytic near z = a , and that the matching P ( a ) ( z ) = ( I + O ( n − 1 )) G ( z ) , n → ∞ , z ∈ ∂ U a . . = D δ ( a ) , (6.23) holds uniformly for s ≥ − s 0 with an y ﬁxed s 0 > 0. 6.8. Conclusion of the asymptotic analysis. W e are ready to conclude the asymptotic analysis. Set Γ R . . = Γ S \ ( U 0 ∪ U a ∪ (0 , a )) , and orient the parts ∂ U 0 and ∂ U 0 of Γ R in the clockwise direction. Let P ( a ) and P (0) b e the local parametrices near z = a and z = 0, respectively , and set P ( z ) . . =      P (0) ( z ) , z ∈ U 0 , P ( a ) ( z ) , z ∈ U a , G ( z ) , z ∈ C \  Γ S ∪ U 0 ∪ U a  . Mak e the transformation R ( z ) . . = S ( z ) P ( z ) − 1 , z ∈ C \ Γ R . Then R satisﬁes the following RHP . RHP 6.13. Find a 2 × 2 matrix-v alued function R : C \ Γ R → C 2 × 2 with the following prop erties. (i) R is analytic on C \ Γ R . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 53 (ii) The jump condition R + ( ζ ) = R − ( ζ ) J R ( ζ ), ζ ∈ Γ A holds, where J R ( z ) . . =        G ( z ) J S ( z ) G ( z ) − 1 , z ∈ Γ R \ ( ∂ U 0 ∪ ∂ U a ) , P (0) ( z ) G ( z ) − 1 , z ∈ ∂ U 0 , P ( a ) ( z ) G ( z ) − 1 , z ∈ ∂ U a . (iii) As z → ∞ , R ( z ) = I + O ( z − 1 ) , where the error term is v alid uniformly for | ζ | > R with some R > 0 indep enden t of n . With standard arguments, in particular thanks to ( 6.19 ) and ( 6.23 ), we obtain the next result. Theorem 6.14. The matrix R satisﬁes R ( z ) = I + O ( n − 1 ) , n → ∞ , uniformly for z ∈ C \ Γ R . F urthermor e, this estimate holds for b oundary values R ± along Γ R , in L ∞ and L 1 norms, uniformly for s ≥ − s 0 with any ﬁxe d s 0 > 0 . This last result concludes the asymptotic analysis, and now w e mov e on to deriving its consequences. 7. Conclusion of main resul ts No w that the asymptotic analysis for orthogonal p olynomials is complete, w e are ready to dra w its consequences, in particular pro ving our main results. F or reference throughout this section, w e trace back all the transformations of the RH analysis for OPs, whic h yields Y ( z ) = e nℓ V σ 3 R ( z ) P ( z ) I + ( χ L + ( z ) − χ L − ( z )) z − α e 2 nϕ ( z ) σ n ( z ) E 21 ! e − n ( ϕ ( z ) − V ( z ) / 2) σ 3 , z ∈ C , (7.1) where we recall that L ± are the lenses used in the transformation T 7→ S (see Figure 2 ), and ϕ is as in ( 6.3 ). 7.1. Asymptotics for the kernel: pro of of Theorem 2.4 . T o obtain the asymptotic b eha vior of the k ernel, w e follo w a standard route in RHPs. How ev er, in the case considered here, the lo cal parametrix Ψ n dep ends on n in a non trivial manner, so for completeness w e decided to presen t the detailed argumen t. Thanks to ( 6.9 ), the correlation kernel ( 2.8 ) can b e expressed in terms of the solution of the RHP 6.2 for OPs as b K n ( x, y | s ) = p ω n ( x | s ) ω n ( y | s ) 2 π i( x − y ) e T 2 Y + ( y ) − 1 Y + ( x ) e 1 . Next, stressing that w e are interested in x, y ∈ (0 , ∞ ) in a neighborho o d of the hard edge, w e plug ( 7.1 ) in to this identit y . Thanks to Theorem 6.14 , standard arguments sho w that R ( y ) − 1 R ( x ) = I + O  x − y n  , 54 L. MOLAG, G. SIL V A, AND L. ZHANG as n → ∞ , uniformly for x, y in compact sets of C (in particular, the uniform conv ergence on C \ Γ R can be extended to compact subsets of C ). Th us, b K n ( x, y ) = ( xy ) α/ 2 p σ n ( x ) p σ n ( y ) e − n ( ϕ ( x )+ ϕ ( y )) 2 π i( x − y ) × " I − y − α e 2 nϕ ( y ) σ n ( y ) E 21 ! P (0) ( y ) − 1  I + O  x − y n  P (0) ( x ) I + x − α e 2 nϕ ( x ) σ n ( x ) E 21 !# 21 , + , v alid uniformly for x, y in compacts of (0 , ∞ ) ∩ U 0 , where w e recall that U 0 is the neigh borho o d where the local parametrix P = P (0) w as constructed in Section 6.6 . Using no w the v ery construction of P (0) from ( 6.16 ) and ( 6.13 ), we simplify this last expression to b K n ( x, y ) = − e π i α p σ n ( x ) p σ n ( y ) 2 π i( x − y ) "  I + e − π i α σ n ( y ) E 21  Ψ n ( ζ ( y )) − 1 σ 3 n − σ 3 / 2 E 1 ( y ) − 1 ×  I + O  x − y n  E 1 ( x ) n σ 3 / 2 σ 3 Ψ n ( ζ ( x ))  I − e − π i α σ n ( x ) E 21  # 21 , + , The function E 1 is analytic near the origin and independent of n . Standard arguments show that E 1 ( y ) − 1 E 1 ( x ) = I + O ( x − y ) . Hence, setting from no w on ζ . . = ζ ( x ) = n 2 ψ ( x ) , ξ . . = ζ ( y ) = n 2 ψ ( y ) , w e update the expression ab ov e to b K n ( x, y ) = − e π i α p σ n ( x ) p σ n ( y ) 2 π i( x − y ) × "  I + e − π i α σ n ( y ) E 21  Ψ n ( ξ ) − 1 ( I + O ( x − y )) Ψ n ( ζ )  I − e − π i α σ n ( x ) E 21  # 21 , − . T o obtain the formula abov e, we mo v ed from a +-b oundary v alue to a − -b oundary v alue in virtue of the change of orientation coming from x 7→ ζ ; see ( 6.6 ). With c V b eing the constant in ( 6.6 ), let us scale x = u c V n 2 , y = v c V n 2 , with u, v > 0, so that, from ( 6.6 ), ζ ( x ) = − u + O ( n − 2 ) and ξ = ζ ( y ) = − v + O ( n − 2 ) . F or u, v in compact subsets of the positive axis, the v alues ζ = ζ ( x ) and ξ = ζ ( y ) remain uniformly in compact subsets of the negative axis, where the mo del problem Ψ n, − is uniformly b ounded in n (see Prop osition 6.11 –(iii)). Recalling also ( 6.14 ), we up date this estimate to 1 c V n 2 b K n  u c V n 2 , v c V n 2  = e π i α p b σ n ( − u ) p b σ n ( − v ) 2 π i( v − u ) × "  I + e − π i α b σ n ( − v ) E 21  Ψ τ ( − v ) − 1 Ψ τ ( − u )  I − e − π i α b σ n ( − u ) E 21  # 21 , − + O ( n − 2 ) . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 55 where Ψ τ = Ψ n and b σ n ( ζ ) = σ n ( x ( ζ )) is admissible and suc h that b σ n ( u ) → σ ∞ ( u ) = (1 + e − h ∞ ( u ) ) − 1 . Recalling that σ ∞ ( ζ ) = (1 + e − h ∞ ( ζ ) ) − 1 and ( 6.14 ), and applying Theorem 5.1 we conclude the pro of of Theorem 2.4 . 7.2. Asymptotics for the multiplicativ e statistics: pro of of Theorem 2.6 . In this section, ∂ denotes the deriv ative with resp ect to the parameter s , whereas ′ will b e used for the deriv ative with respect to the sp ectral (the RHP) v ariable, so that for instance ∂ σ n ( x | s ) = ∂ σ n ∂ s ( x | s ) and σ ′ n ( x | s ) = ∂ σ n ∂ x ( x | s ) . Let us c hoose the neigh b orho o ds U 0 and U a where the local parametrices w ere constructed so that, to simplify notation, U 0 ∩ (0 , ∞ ) = (0 , δ ) and U a ∩ (0 , ∞ ) = ( a − δ, a + δ ) . W e split ( 6.10 ) as log L Q n ( s ) = − 1 2 π i Z ∞ s ( J h ( u ) + J b ( u ) + J s ( u ) + J g ( u )) d u, (7.2) where the terms J h , J b , J s , J g corresp ond to the x -in tegrals o v er the neighborho o d (0 , δ ) of the hard edge region, ov er the bulk region ( δ, a − δ ), ov er the neighborho o d ( a − δ, a + δ ) of the soft edge, and ov er the gap interv al ( a + δ, ∞ ) aw a y from the supp ort of µ V , resp ectively . W e split the analysis of each suc h term in diﬀeren t sections b elow. 7.2.1. Contributions ne ar the har d e dge. In this section w e analyze the in tegral Z ∞ s J h ( u )d u, with J h ( s ) . . = Z δ 0  Y ( x | s ) − 1 Y ′ ( x | s )  21 , + ω n ( x | s ) ∂ log σ n ( x | s )d x, (7.3) whic h will turn out to giv e the leading con tribution to ( 7.2 ). The idea is standard: w e un wrap the transformations Y 7→ · · · 7→ R of the RHP analysis, with careful considerations along the wa y . When doing so, we will ultimately appro ximate Y by the lo cal parametrix P (0) near the hard edge, and this parametrix will giv e us the leading contribution. F or x ∈ (0 , δ ], that is, in the neighborho o d where w e constructed the hard edge parametrix, we start from ( 7.1 ) and use ( 6.13 ) and ( 6.16 ) to write Y + ( x ) = e nℓ V σ 3 R + ( x ) E n ( x ) b Ψ n, + ( x )  I + e − π i α E 21 σ n ( x )  x − α σ 3 / 2 e nV ( x ) σ 3 / 2 e π i α σ 3 / 2 , where E n and b Ψ n are as in ( 6.16 ) and ( 6.15 ). With the notation ∆ x , ∆ ζ in tro duced in ( 2.21 )–( 2.22 ) in mind, w e compute [ Y ( x ) − 1 Y ′ ( x )] 21 , + = − e π i α + nV ( x ) x − α  ∆ x  Ψ n ( ζ ( x ))  I − e − π i α σ n ( x ) E 21  21 , + − e π i α + nV ( x ) x − α  I + e − π i α σ n ( x ) E 21  R h ( x )  I − e − π i α σ n ( x ) E 21  21 , + , whic h is v alid for x ∈ (0 , δ ) and where w e hav e set R h ( x ) = R h ( x | s ) . . = Ψ n ( ζ ( x )) − 1 σ 3 n − σ 3 / 2  ∆ x E 1 ( x ) + E 1 ( x ) − 1 ∆ x R ( x ) E 1 ( x )  n σ 3 / 2 σ 3 Ψ n ( ζ ( x )) . 56 L. MOLAG, G. SIL V A, AND L. ZHANG Plugging the result in to ( 7.3 ), w e obtain the iden tit y J h ( u ) = − e π i α Z δ 0 ζ ′ ( x )  ∆ ζ  Ψ n ( ζ )  I − e − π i α σ n ( x ( ζ ) | u ) E 21  21 , + ∂ σ n ( x | u )d x − e π i α Z δ 0  I + e − π i α σ n ( x | u ) E 21  R h , + ( x )  I − e − π i α σ n ( x | u ) E 21  21 ∂ σ n ( x | u )d x. (7.4) W e no w obtain a b ound for the in tegral on the second line. F rom Proposition 6.11 –(i) and Theorem 6.14 w e estimate n − σ 3 / 2 ∆ x E 1 ( x ) n σ 3 / 2 = O ( n ) , and n − σ 3 / 2 E 1 ( x ) − 1 ∆ x R ( x ) E 1 ( x ) n σ 3 / 2 = O (1) , as n → ∞ , so that R h ( x ) = Ψ n ( ζ ( x )) − 1 O ( n ) Ψ n ( ζ ( x )) , n → ∞ , (7.5) where the error term is v alid uniformly for x ∈ (0 , δ ) and uniformly for s ≥ − s 0 with an y ﬁxed s 0 > 0. T o estimate the withstanding term Ψ n , w e explore the c hange of v ariables ζ = n 2 ψ ( x ), whic h maps the interv al [0 , δ ] to an in terv al of the form [ − R n , 0], with R n = O ( n 2 ). W e split such interv al in three pieces [ − R n , 0] = [ − R n , − M ] ∪ [ − M , − 1 / M ] ∪ [ − 1 / M , 0] , where M > 0 is independent of n and c hosen so that Proposition 6.11 –(iii),(iv),(v) are all v alid. In what follo ws, recall also that 1 /σ n ( z ) = 1 + e h n ( ζ ) , with h n = h τ admissible in the sense of Deﬁnition 4.1 . In the compact interv al [ − M , − 1 / M ] the con vergence e − h n ( ζ ) → e − s +( − 1) m +1 u ζ m tak es place uni- formly , and this limit is b ounded in this same interv al, uniformly also for s ≥ − s 0 . Combining with Prop osition 6.11 –(iii) and ( 7.5 ), we obtain  I + e − π i α σ n ( x ) E 21  R h ( x )  I − e − π i α σ n ( x ) E 21  21 , + = O ( n ) , n → ∞ , (7.6) v alid uniformly for ζ ( x ) ∈ [ − M , − 1 / M ] and uniformly for s ≥ − s 0 . On the interv al [ − 1 / M , 0], we use ( 6.18 ) and Prop osition 6.11 –(iv) and write Ψ n, + ( ζ ( x ))  I − e − π i α σ n ( x ) E 21  = O (1) ζ α 2 σ 3 − ( I + a − ( ζ ) E 12 ) σ ( x ) σ 3 / 2 , n → ∞ , and therefore using again ( 7.5 ),  I + e − π i α σ n ( x ) E 21  R h ( x )  I − e − π i α σ n ( x ) E 21  21 , + = O  n 1+2 α | x | α  , n → ∞ , (7.7) v alid uniformly for ζ = ζ ( x ) ∈ [ − 1 / M , 0] and s ≥ − s 0 , and where w e used that ζ = n 2 ψ ( x ), ψ is conformal near the origin, and σ n is bounded along the real axis. Finally , on the in terv al [ − R n , − M ] we now use Prop osition 6.11 –(v) and conclude that Ψ n, + ( ζ ( x ))  I − e − π i α σ n ( x ) E 21  = O (1) ζ − 1 4 σ 3 − U 0 ( I − (1 + e − h n ( ζ ) ) e − π i α − 4i | ζ | 1 / 2 E 21 ) e − 2i | ζ | 1 / 2 σ 3 , ζ ( x ) ∈ [ − R n , − M ] , n → ∞ , also with uniform error term for s ≥ − s 0 , and emphasizing that this error term dep ends on the ﬁxed v alue M but it is indep enden t of R n . As said b efore, h n is admissible, and Deﬁnition 4.1 –(iii) implies CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 57 in particular that (1 + e − h n ) is b ounded along the negativ e axis, uniformly for s ≥ − s 0 and uniformly in n as w ell. With this observ ation in mind, the last estimate and ( 7.5 ) together imply  I + e − π i α σ n ( x ) E 21  R h ( x )  I − e − π i α σ n ( x ) E 21  21 , + = O ( n 2 (1 + | x | 1 / 2 )) , n → ∞ , (7.8) uniformly for ζ = ζ ( x ) ∈ [ − R n , − M ] and s ≥ − s 0 . In fact, x is bounded in the corresponding set, and the error term abov e is eﬀectiv ely O ( n 2 ). Again having in mind that ζ = ζ ( x ) = n 2 ψ ( x ) and that ψ ( x ) is an n -indep endent conformal map near the origin, w e use ( 7.6 ),( 7.7 ) and ( 7.8 ), obtaining that there exists a constan t C > 0 for which the in tegral on the second line of ( 7.4 ) is bounded by C n n 2 α Z ζ − 1 ( − 1 / M ) 0 ∂ σ n ( x | u ) x α d x + Z ζ − 1 ( M ) ζ − 1 ( − 1 / M ) ∂ σ n ( x | u )d x + n Z δ ζ − 1 ( M ) ∂ σ n ( x | u )d x ! , where the additional factor n in the last term comes from the factor ζ σ 3 / 4 = ( n 2 ψ ( x )) σ 3 / 4 in ( 7.8 ). An explicit calculation shows that 0 ≤ ∂ σ n ( x | s ) = e − s − n 2 m Q ( x ) (1 + e − s − n 2 m Q ( x ) ) 2 ≤ e − s − n 2 m Q ( x ) . Ha ving in mind that M > 0 is ﬁxed and R n = O ( n 2 ), w e obtain that ( 7.8 ) - and thus the second line in ( 7.4 ) - is b ounded b y C n e − u n 2 α Z 1 /η 0 x α e − n 2 m Q ( x ) d x + n Z η 1 /η e − n 2 m Q ( x ) d x ! for some η , C > 0, which w e emphasize may depend on s 0 > 0 ﬁxed but it is indep endent of s ≥ − s 0 . Under our conditions on Q (recall Assumption 2.2 ), it is immediate that n Z ∞ 1 /η e − n 2 m Q ( x ) d x = O (e − η ′ n 2 m ) , for some η ′ > 0 indep enden t of s , and the change of v ariables n 2 x = y yields the estimate n 2 α Z 1 /η 0 x α e − n 2 m Q ( x ) d x = O ( n − 2 ) . Using these estimates in ( 7.4 ), w e ﬁnally arrive at the estimate J h ( u ) = − e π i α Z δ 0 ζ ′ ( x )  ∆ ζ  Ψ n ( ζ )  I − e − π i α σ n ( x ( ζ ) | u ) E 21  21 , + ∂ σ n ( x | u )d x + O  e − u n  , v alid as n → ∞ , uniformly for u ≥ − s 0 , for an y s 0 > 0. W e no w c hange v ariables x 7→ ζ in the integral. F or that, notice that σ n ( x ( ζ ) | u ) = 1 1 + e − h n ( ζ | u ) , where h n = h τ is admissible in the sense of Deﬁnition 4.1 . With R = R n determined b y relation − n 2 R n = ζ ( δ ) , w e kno w that R n > 0 is bounded from ab ov e, and also from b elow aw a y 0, and this estimate becomes J h ( u ) = e π i α Z 0 − n 2 R h ∆ ζ h Ψ n ( ζ )  I − e − π i α  1 + e − h n ( ζ | u )  E 21 ii 21 , − e − h n ( ζ | u ) (1 + e − h n ( ζ | u ) ) 2 d ζ + O  e − u n  , 58 L. MOLAG, G. SIL V A, AND L. ZHANG v alid as n → ∞ , uniformly for u ≥ − s 0 , for any s 0 > 0 ﬁxed. Using no w Theorem 5.2 , we obtain Z ∞ s J h ( u )d u = 2 π i Z ∞ s Z 0 −∞ K α ( − ζ , − ζ | u ) e − h ∞ ( ζ | u ) 1 + e − h ∞ ( ζ | u ) d ζ d u + O  e − s n  , n → ∞ , (7.9) uniformly for s ≥ s 0 . 7.2.2. Contributions away fr om the supp ort. Returning to the analysis of ( 7.2 ), w e now estimate Z ∞ s J g ( u )d u, with J g ( s ) . . = Z ∞ a + δ [ ∆ x Y ( x | s )] 21 , + ω n ( x | s ) ∂ log σ n ( x | s )d x. In the interv al [ a + δ, ∞ ), the unwrap of the transformations yields Y ( z ) = e nℓ V σ 3 R ( z ) G ( z ) e − n ( ϕ ( x ) − V ( x ) / 2) σ 3 . Therefore, using the explicit expression for G in ( 6.11 ) we obtain [ ∆ x Y ( x )] 21 , + = e − 2 nϕ ( x )+ nV ( x )+2 p ( x ) D ( x ) 2  ∆ x M ( z ) + M ( x ) − 1 D σ 3 ∞ e p 0 σ 3 ∆ x R + ( x ) e − p 0 D − σ 3 ∞ M ( x )  21 . A direct calculation from ( 6.12 ) rev eals that [ ∆ x M ( x )] 21 = i 4 g ′ ( x ) g ( x ) = i 4 a x ( x − a ) , whic h is clearly b ounded on [ a + δ, ∞ ). Likewise, all the entries of M are b ounded in this same interv al, b ecause so is g . The term D is discussed after ( 6.11 ), and it is con tinuous and nonzero on the in terv al [ a + δ, ∞ ), and conv erges to D ∞  = 0 at ∞ . Therefore the factor D ( x ) − 2 is also bounded on [ a + δ, ∞ ). The terms D ∞ and p 0 are constant in x , with the former b eing indep endent of n as well while the latter is uniformly b ounded in n thanks to Prop osition 6.6 . Finally , thanks to Theorem 6.14 w e kno w that the factor ∆ x R + is O ( n − 1 ). All in all, we conclude that, with the exception of the exp onential comp onent, all terms on the right hand side of ( 7.2.2 ) are b ounded, and therefore for some M > 0,    [ ∆ x Y ( x | s )] 21 , + ω n ( x | s ) ∂ log σ n ( x | s )    ≤ e − s − n 2 m Q ( x ) (1 + e − s − n 2 m Q ( x ) ) 2 x α e − 2 nϕ ( x )+2 p ( x ) , x ≥ a + δ, where we used the explicit expressions for ω n and σ n from ( 2.2 ) and ( 2.3 ). The function p do es dep end on b oth n and s , but thanks to Prop osition 6.6 it remains b ounded uniformly b oth for s ≥ − s 0 and x ≥ a + δ as n → ∞ . Therefore, for a new constan t M > 0 the right-hand side abov e is b ounded b y M x α e − s − n 2 m Q ( x ) − 2 nϕ ( x ) . W e in tegrate this b ound and use that Q ( x ) > 0 on the positive axis, concluding that     Z ∞ s J g ( u )d u     ≤ M e − s Z ∞ a + δ x α e − 2 nϕ ( x ) d x. The function ϕ is given in ( 6.3 ), and as observ ed from ( 6.7 ) it gro ws p olynomially at ∞ . A simple estimate then shows that this remaining in tegral is O (e − η n ), for some η > 0. W e just concluded that Z ∞ s J g ( u )d u = O (e − s − η n ) , n → ∞ , (7.10) uniformly for s ≥ − s 0 . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 59 7.2.3. Contributions ne ar the soft e dge. W e no w estimate Z ∞ s J s ( u )d u, with J s ( s ) . . = Z a + δ a − δ [ ∆ x Y ( x | s )] 21 , + ω n ( x | s ) ∂ log σ n ( x | s )d x. (7.11) Let χ 0 = χ (0 ,a ) b e the characteristic function of the interv al (0 , a ). Recalling ( 7.1 ), near the soft edge at z = a , the unwrap of the transformations giv es the relation Y + ( x ) = e nℓ V σ 3 R ( x ) P + ( x ) I + e 2 nϕ + ( x ) x α σ n ( x ) χ 0 ( x ) E 21 ! e − n (2 ϕ + ( x ) − V ( x )) σ 3 / 2 , a − δ ≤ x ≤ a + δ, where P = P ( a ) for the rest of this section. Therefore, in the same interv al, [ ∆ x Y ( x )] 21 , + = e − n (2 ϕ + ( x ) − V ( x )) e 2 nϕ + ( x ) x α σ n ( x ) ! ′ χ 0 ( x ) + e − n (2 ϕ + ( x ) − V ( x )) " I − χ 0 ( x ) e 2 nϕ + ( x ) x α σ n ( x ) E 21 ! ×  ∆ x P + ( x ) + P + ( x ) − 1 ∆ x R ( x ) P + ( x )  I + χ 0 ( x ) e 2 nϕ + ( x ) x α σ n ( x ) E 21 ! # 21 . (7.12) Using ( 6.21 ) and ( 6.22 ), as w ell as Theorem 6.14 and the analyticit y of E ( a ) 1 , w e write ∆ x P + ( x ) + P + ( x ) − 1 ∆ x R ( x ) P + ( x ) = ∆ x h ( σ n ( x ) x α ) − σ 3 / 2 e n e ϕ + ( x ) σ 3 i + ( σ n ( x ) x α ) σ 3 / 2 e − n e ϕ + ( x ) σ 3 × h ∆ x  A + ( n 2 / 3 e ψ ( x ))  + A + ( n 2 / 3 e ψ ( x )) − 1 O ( n 1 / 3 ) A + ( n 2 / 3 e ψ ( x )) i e n e ϕ + ( x ) σ 3 ( σ n ( x ) x α ) − σ 3 / 2 . T o bound the terms in v olving A w e pro ceed in a w a y similar to what w e did near z = 0. F or a n um b er R > 0 to b e tak en suﬃciently large but ﬁxed, deﬁne y n = y n ( R ) , x n = x n ( R ) ∈ ( a − δ, a + δ ) through the relation [ y n , x n ] = { x ∈ [ a − δ, a + δ ] | n 2 / 3 | e ψ ( x ) | ≤ R } . Cho ose R > 0 in such a wa y that the expansion ( 6.20 ) holds for | ζ | ≥ R . This is done in such a wa y that, on [ a − δ, a + δ ] \ [ y n , x n ] w e ma y use this expansion ( 6.20 ) to obtain b ounds for A , whereas along [ y n , x n ] w e may simply use that A remains b ounded. F ollowing this idea, we obtain the same b ound in b oth in terv als, namely ∆ x P + ( x ) + P + ( x ) − 1 ∆ x R ( x ) P + ( x ) = O ( n ) , x ∈ [ a − δ, a + δ ] , where w e remark that we used that σ n ( x ) x α remains b ounded in an y small neighborho o d of x = a and the iden tit y n e ϕ = 2 3 ( n 2 / 3 e ψ ) 3 / 2 . W e use this last estimate to bound the term in the right-most side of ( 7.12 ). In addition, we b ound the deriv ative term m ultiplying ξ 0 b y observing that ϕ + is purely imaginary on (0 , a ). As a result, w e obtain the ov erall b ound [ ∆ x Y ( x )] 21 , + = O  n e − n (2 ϕ + ( x ) − V ( x ))  , n → ∞ , (7.13) v alid uniformly for | x − a | ≤ δ and s ≥ s 0 . Using this estimate in ( 7.11 ), w e obtain that J s ( s ) is b ounded b y a uniform constan t times n Z a + δ a − δ x α e − s − n 2 m Q ( x ) (1 + e − s − n 2 m Q ( x ) ) 2 e − nϕ + ( x ) d x ≤ n e − s Z a + δ a − δ x α e − s − n 2 m Q ( x ) e − nϕ + ( x ) d x. 60 L. MOLAG, G. SIL V A, AND L. ZHANG In the remaining in tegral, b oth functions Q ( x ) and ϕ + ( x ) are contin uous on the in terv al of in tegration, with Q b eing strictly p ositiv e there. In particular, this observ ation implies that the whole integral is O (e − η n 2 m ), for some η > 0. In tegrating this result no w in the s -v ariable, we obtain the estimate Z ∞ s J s ( u )d u = O  e − s − η n 2 m  , n → ∞ , (7.14) for some (new) v alue η > 0, uniformly for s ≥ s 0 . 7.2.4. Contributions fr om the bulk. T o ﬁnalize the analysis of the terms in ( 7.2 ), w e now analyze Z ∞ s J b ( u )d u, with J b ( s ) . . = Z a − δ δ [ ∆ x Y ( x | s )] 21 , + ω n ( x | s ) ∂ log σ n ( x | s )d x. F or δ < x < a − δ , the un wrap of the transformations no w unrav els as Y + ( x ) = e nℓ V σ 3 R + ( x ) G + ( z ) I + x − α e 2 nϕ + ( x ) σ n ( z ) E 21 ! e − n ( ϕ + ( z ) − V ( x ) / 2) σ 3 , see ( 7.1 ). The function ϕ + is purely imaginary along (0 , a ). The function R , as w ell as its deriv ativ e, remains b ounded as n → ∞ . As it can b e seen from Prop osition 6.7 , the global parametrix (as w ell as its deriv ative) remains b ounded as n → ∞ . This means that, when applying the identit y ab o v e to esti- mate ∆ x Y + , p ossibly gro wing terms come only from the exp onen tial part e − n ( π + ( x ) − V ( x ) / 2) σ 3 and its deriv ative. With this observ ation in mind, we see immediately that [ ∆ x Y ( x )] 21 , + = O ( n e nV ( x ) ) , n → ∞ , uniformly for x ∈ ( δ , a − δ ) and uniformly for s ≥ s 0 . This estimate is not necessarily optimal, but suﬃcien t for our purp oses. Pro ceeding in exactly the same w a y as we did from ( 7.13 ) on wards, we obtain Z ∞ s J b ( u )d u = O  e − s − η n 2 m  , n → ∞ , (7.15) for some η > 0, uniformly for s ≥ s 0 . 7.2.5. Summary of c ontributions and c onclusion of the pr o of of The or em 2.6 . Summarizing, w e combine ( 7.9 ), ( 7.10 ), ( 7.14 ) and ( 7.15 ) into ( 7.2 ), obtaining log L Q n ( s ) = − Z ∞ s Z 0 −∞ K α ( − ζ , − ζ | u ) e − h ∞ ( ζ | u ) 1 + e − h ∞ ( ζ | u ) d ζ d u + O  e − s n  , n → ∞ , where the error term is uniform for s ≥ s 0 with an y ﬁxed s 0 ∈ R . The pro of of Theorem 2.6 is completed b y making the c hange of v ariables ζ 7→ − ζ and using that (1 + e − h ∞ ( ζ | s ) ) − 1 = σ Φ  4 x 2 ζ | s  . Appendix A. The Bessel p arametrix The Bessel parametrix Φ (Bes) α is a solution to a canonical RHP whic h dep ends parametrically on α > − 1, and which w as ﬁrst introduced in [ 40 ] and since then has b een used widely as a mo del problem in the asymptotic analysis of RHPs. Set Γ ± . . = (e ± 2 π i / 3 ∞ , 0], Γ 0 . . = ( −∞ , 0] and Γ . . = Γ 0 ∪ Γ + ∪ Γ − , which is consisten t with ( 3.1 ) for the c hoice m = 1. Introduce CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 61 Φ (Bes) α ( z ) . . = I α (2 z 1 / 2 ) i π K α (2 z 1 / 2 ) 2 π i z 1 / 2 I ′ α (2 z 1 / 2 ) − 2 z 1 / 2 K ′ α (2 z 1 / 2 ) !      I , | arg z | < 2 π 3 , ( I ∓ e ± π i α E 21 ) , 2 π 3 < ± arg z < π , (A.1) where I α ( z ) and K α ( z ) denote the mo diﬁed Bessel functions and the principal branc h is taken for z 1 / 2 . F ollowing [ 40 , Pages 365–368, in particular Theorem 6.3, and also Equation (8.4)], the matrix Φ (Bes) α is the solution to the follo wing RHP . RHP A.1. (i) Φ (Bes) α ( z ) is deﬁned and analytic in C \ ( −∞ , 0]. (ii) F or z ∈ Γ \ { 0 } , w e ha v e Φ (Bes) α, + ( z ) = Φ (Bes) α, − ( z ) J (Bes) ( z ) , with J (Bes) ( z ) . . = ( I + e ± π i α E 21 , z ∈ Γ ± , E 12 − E 21 , z ∈ Γ 0 . (iii) With ( α, 0) := 1, ( α, k ) . . = (4 α 2 − 1)(4 α 2 − 3 2 ) · · · (4 α 2 − (2 k − 1) 2 ) 2 2 k k ! , k ∈ Z > 0 , and a k ( α ) . . = ( α, k − 1) 4 k k  α 2 + k 2 − 1 4  , b k ( α ) . . = ( α, k − 1) 4 k  k − 1 2  , k ∈ Z ≥ 0 , (A.2) the function Φ (Bes) α admits the following asymptotic expansion as z → ∞ , Φ (Bes) α ( z ) ∼ (2 π ) − σ 3 / 2 z − σ 3 / 4 U 0 I + ∞ X k =1 ( − 1) k z k/ 2 ( a k ( α ) I + b k ( α ) σ 2 ) σ k 3 ! e 2 z 1 / 2 σ 3 , (A.3) where σ 2 =  0 − i i 0  and w e choose the principal branc hes of z 1 / 4 and z 1 / 2 . (iv) As z → 0 w e ha v e Φ (Bes) α ( z ) = Φ (Bes) α, 0 ( z ) z α σ 3 / 2 ( I + a ( z ) E 12 ) ( I − ( χ S + ( ζ ) e π i α − χ S − ( ζ ) e − π i α ) E 21 ) , (A.4) where Φ (Bes) α, 0 ( z ) is analytic in a neigh b orho o d of the origin, we c ho ose principal branch for z α/ 2 , and a ( z ) = a α ( z ) is as in ( 3.7 ). The lo cal behavior in ( A.4 ) is sligh tly more precise than the one describ ed in [ 40 , Equations (6.19)– (6.21)]. Nevertheless, applying Prop osition 3.2 with the choice σ ≡ 1, w e get that the b eha vior from [ 40 ] implies ( A.4 ). Since w e also need the v alue Φ (Bes) α, 0 (0) at the origin, we no w v erify the b ehavior ( A.4 ) directly from ( A.1 ). The modiﬁed Bessel function I α ( w ) is deﬁned through [ 32 , (10.25.2)] I α ( w ) . . =  w 2  α F α ( w 2 / 4) , where F α ( w ) . . = ∞ X k =0 w k k !Γ( α + k + 1) . Observ e that F α is an entire function for any α ∈ R . F rom this expression for I α , w e compute I α (2 z 1 / 2 ) = z α/ 2 F α ( z ) and z 1 / 2 I ′ α (2 z 1 / 2 ) = z α/ 2  α 2 F α ( z ) + z F ′ α ( z )  . (A.5) 62 L. MOLAG, G. SIL V A, AND L. ZHANG Using there relations and ( A.1 ) and ( A.4 ), w e are able to compute the matrix function Φ (Bes) α, 0 explic- itly . F or an y α > − 1, w e compute that for z > 0 the ﬁrst column of Φ (Bes) α, 0 is giv en by  Φ (Bes) α, 0 ( z )  11 = z − α/ 2  Φ (Bes) α ( z )  11 = F α ( z ) = z − α/ 2 I α (2 z 1 / 2 ) (A.6) and  Φ (Bes) α, 0 ( z )  21 = z − α/ 2  Φ (Bes) α ( z )  21 = π i αF α ( z ) + 2 π i z F ′ α ( z ) = 2 π i z (1 − α ) / 2 I ′ α (2 z 1 / 2 ) . (A.7) By analytic contin uation, these iden tities extend from z > 0 to z ∈ C . The second column of Φ (Bes) α, 0 could b e describ ed similarly , but w e will not need it so we skip the details. As a consequence, it is readily seen from ( A.1 ) and ( A.4 ) that Φ (Bes) α, 0 (0) =            1 Γ( α +1) i Γ( α ) 2 π π i Γ( α ) Γ( α +1) 2 ! , α  = 0 ,  1 − i π log 2 0 1  , α = 0 . (A.8) F or us, w e need Φ (Bes) α to construct the particular solution Ψ ( · | h = + ∞ ) = Ψ (Bes) α of the mo del problem RHP 3.1 . Setting Ψ (Bes) α ( z ) . . = ( I + i( a 1 ( α ) + b 1 ( α )) E 21 )(2 π ) σ 3 / 2 Φ (Bes) α ( z ) , where a 1 ( α ) and b 1 ( α ) are giv en in ( A.2 ), the conditions in RHP A.1 imply that indeed Ψ (Bes) α = Ψ ( · | h = + ∞ ). F or the record, notice that  Ψ (Bes) α ( z )  j 1 = (2 π ) ( − 1) j +1 / 2  Φ (Bes) α ( z )  j 1 , j = 1 , 2 , (A.9) and w e also state that ( A.3 ) yields Ψ (Bes) α ( z ) ∼ I + ∞ X k =1 1 z k Ψ (Bes) ∞ ,k ( α ) ! z − σ 3 / 4 U 0 e 2 z 1 / 2 σ 3 , z → ∞ , (A.10) with Ψ (Bes) ∞ ,k ( α ) . . = ( I + i( a 1 ( α ) + b 1 ( α )) E 21 )  a 2 k ( α ) − b 2 k ( α ) i( a 2 k − 1 ( α ) − b 2 k − 1 ( α )) − i( a 2 k +1 ( α ) + b 2 k +1 ( α )) a 2 k ( α ) + b 2 k ( α )  , k ≥ 1 . In particular,  Ψ (Bes) α ( z )  11 = z − 1 / 4 √ 2  1 + O ( z − 1 )  e 2 z 1 / 2 ,  Ψ (Bes) α ( z )  21 = i z 1 / 4 √ 2  1 + O ( z − 1 )  e 2 z 1 / 2 , z → ∞ , and  Ψ (Bes) ∞ , 1 ( α )  12 = i(4 α 2 − 1) 16 . (A.11) F urthermore, the lo cal expansion ( 3.9 ) holds with, as said, σ ( z ) ≡ 1 and Ψ 0 ( z ) = Ψ (Bes) 0 ,α ( z ) . . = ( I + i( a 1 ( α ) + b 1 ( α )) E 21 )(2 π ) σ 3 / 2 Φ (Bes) α, 0 ( z ) . CONDITIONAL THINNING AND MUL TIPLICA TIVE ST A TS OF LU-TYPE ENSEMBLES 63 Appendix B. Asymptotic anal ysis for a class of integrals Consider a function q holomorphic on a neigh b orho o d of the origin, with expansion q ( z ) = q 0 z m (1 + O ( z )) , z → 0 , for some m ≥ 1 and q 0 > 0. Ass ume in addition that q > 0 on an interv al of the form [0 , δ ], δ > 0. F or a n um ber β > − 1 and a smooth function f on [0 , δ ] with f (0)  = 0, in troduce I t ( s ) . . = Z δ 0 x β f ( x ) log(1 + e − s − tq ( x ) )d x, t > 0 , s ∈ R . W e assume that x β f ( x ) ∈ L 1 [0 , δ ]. W e are interested in the b ehavior of I t ( s ) when t → ∞ sim ultaneously with s → + ∞ . F or the next result, set F β ( s ) . . = Z ∞ 0 x β log(1 + e − s − x )d x, s ∈ R , β > − 1 . Prop osition B.1. (i) F or s > 0 and any β > − 1 , the identity F β ( s ) = − β Γ( β ) Li β +2 ( − e − s ) holds, wher e Li β stands for the p olylo garithms. F urthermor e, for any s 0 > 0 , the estimate F β ( s ) = β Γ( β ) e − s + O (e − 2 s ) , s → + ∞ , is valid uniformly for s ≥ s 0 . (ii) F or any s 0 ∈ R , the estimate I t ( s ) = f (0) mq β +1 m 0 1 t β +1 m  F β +1 m − 1 ( s ) + O  1 t  , t → ∞ , holds uniformly for s ≥ s 0 . Notice that the function s 7→ F β ( s ) is b ounded on any in terv al of the form [ M , ∞ ), so the estimate (ii) indeed singles out the leading term in I t ( s ) on the range s ∈ ( s 0 , + ∞ ). Pr o of. F or the identit y b etw een F β and the p olylog, w e use the assumption s > 0 to e xpand the log in series and interc hange the order of summation and integration, obtaining F β ( s ) = − ∞ X k =1 ( − e − s ) k k Z ∞ 0 x β e − kx d x = − Γ( β + 1) ∞ X k =1 ( − e − s ) k k β +2 , where the last identit y follo ws performing the c hange of v ariables k x = u in the in tegration. This last series is precisely the series expansion of Li β +2 ( − e − s ), and it also yields the claimed asymptotic estimate for F β . T o pro v e (ii), we start with the simpler case when f ( x ) ≡ 1 and q ( x ) ≡ x . In this case, we c hange v ariables tx = v and obtain Z δ 0 x β log(1 + e − s − tx )d x = 1 t β +1 F β ( s ) − 1 t β +1 Z ∞ δ t v β log(1 + e − s − v )d v . T o estimate the last integral, we use the inequality log(1 + u ) ≤ u , which is v alid for an y u ≥ 0, and obtain Z ∞ δ t v β log(1 + e − s − v )d v ≤ e − s 0 Z ∞ tδ v β e − v d v ≤ e − s 0 Z ∞ 0 v β e − v d v . 64 L. MOLAG, G. SIL V A, AND L. ZHANG The last integral ab o ve is independent of t , and shows that Z δ 0 x β log(1 + e − s − tx )d x = 1 t β +1 F β ( s ) + O (e − η t ) . (B.1) F or the general case, we ﬁx a p ositiv e v alue δ ′ < δ for whic h q is a smo oth bijection on [0 , δ ′ ] and f is smooth on [0 , δ ′ ], and split the integral I t ( s ) = Z δ ′ 0 x β f ( x ) log(1 + e − s − tq ( x ) )d x + Z δ δ ′ x β f ( x ) log(1 + e − s − tq ( x ) )d x. Since q is strictly p ositive on [ δ ′ , δ ], it has a minimum therein, sa y q ( x ) ≥ q 1 > 0. Pro ceeding as b efore, it is immediate to see that the last integral deca ys exp onen tially fast in t , uniformly for s ≥ s 0 , and therefore I t ( s ) = Z δ ′ 0 x β f ( x ) log(1 + e − s − tq ( x ) )d x + O (e − η t ) , t → ∞ , where η > 0 is uniform for s ≥ s 0 . F or the last integral, w e c hange v ariables tq ( x ) = u , and with e q . . = q − 1 b eing the functional in v erse of q on [0 , δ ′ ] w e obtain Z δ ′ 0 x β f ( x ) log(1 + e − s − tq ( x ) )d x = Z q ( δ ′ ) 0 e q ( u ) β f ( e q ( u )) ( e q ) ′ ( u ) log(1 + e − s − tu )d u. A local analysis near the origin shows that e q ( u ) β f ( e q ( u )) ( e q ) ′ ( u ) = f (0) mq ( β +1) /m 0 u ( β +1 − m ) /m + R ( u ) where the error term R satisﬁes | R ( u ) | ≤ R 0 u ( β +1) /m , 0 ≤ u ≤ δ ′ , for some p ositive constan t R 0 that dep ends on f , e q , β but it is indep endent of t, s . 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Conditional thinning and multiplicative statistics of Laguerre-type orthogonal polynomial ensembles

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