Normal matrix models, dbar-problem, and orthogonal polynomials on the complex plane

NORMAL MA TRIX MODELS, ¯ ∂ -PR OBLEM, AND OR THOGONAL POL YNOMI AL S ON THE COMPLEX PLANE ALEXANDER R. ITS AND LEON A. T AKHT AJAN Abstra ct. W e introduce a ¯ ∂ -form ulation of the orthogonal p olynomi- als on the co mplex plane, and hence of the rela ted n ormal matrix model, whic h is exp ected to pla y the same role as the Riemann-Hilb ert for- malism in the theory of orthogonal p olynomials on the line and for the related Hermitian mo del. W e prop ose an analog o f D eift-K riecherbauer- McLaughlin-V enakides-Zh ou asymptotic metho d for the analysis of the relev an t ¯ ∂ -problem, and indicate ho w familiar steps for the Hermitian mod el, e.g. the g -function “undressing”, might lo ok lik e in the case of the normal mo del. W e use th e particular mo del considered recently by P . Elbau and G. F elder as a case stu dy . 1. Introduction In these notes w e attempt to d ev elop for th e norm al matrix mo del a formalism analogous to the Riemann-Hilbert method in the t heory of Her- mitian mat rix mo del. As in the latter case, th e starting p oint is prop er analytical c haracterization of th e relev an t orthogonal p olynomials. Unlik e the Hermitian matrix mo d el, t he orthogonalit y condition for the p olynomials asso ciated with the normal mo d el is form ulated w ith resp ect to a measure on the plane. This, as we will see b elow, leads to the r eplacemen t of the Riemann-Hilb ert pr oblem of [4] by a certain ¯ ∂ -problem. W e shall p resen t in d etail the setting of the ¯ ∂ -problem for the case of wh at we will call in these notes th e Elb au-F elder mo del . This mo d el arises as a natural regu- larizatio n of the norm al matrix mo del of P . Wiegmann and A. Zabr o din in [7, 8] b y restricting the matrix integ ral of the latter to norm al matrices whose eigen v alues lie in a compact domain D of the complex plane. Using the Elbau-F elder mo del as a case study , we shall also outline a p ossible ¯ ∂ - v ersion of the Deift-Kriec herbauer-McLaughlin-V enakides-Zhou (DKMVZ) asymptotic mo d el. The DKMVZ metho d p ro ve d to b e ve ry efficient in the asymptotic analysis of the oscilla tory Riemann-Hilb ert p roblems app earing in Hermitian matrix mo d el. W e h av e n ot y et succeeded in p ro viding com- plete generalizatio n of the DKMVZ scheme for the orthogonal p olynomials on the plane; in fact, we rather highlight ed the chall enging difficulties to b e ov er come. W e hop e, ho wev er, that these notes migh t stimulate f urther dev elopmen t of the analog DKMVZ asymptotic metho d for the orthogonal p olynomials on the plane and related normal matrix mo dels. 1 2 ALEXANDER R . ITS AND LEON A. T AKHT AJAN 2. Preliminaries 2.1. Normal mat rix mo dels and orthogonal p olynomials. Let D b e a b ounded domain on th e complex plane C con taining the origin, and let V ( z ) b e a r eal-v alued s m o oth fu nction on C . F ollo wing P . Elbau and G. F elder [3], we sh all study th e normal matrtix mo d el c haracterized by the partition function Z N defined b y the follo wing N -fold in tegral, Z N = Z · · · Z D N Y i 6 = j | z i − z j | 2 e − N P N k =1 V ( z k ) d 2 z 1 · · · d 2 z N . Let χ D b e the c haracteristic fu nction of the d omain D . Definition 1. Orthogonal p olynomials on C with resp ect to th e measure e − N V ( z ) χ D ( z ) d 2 z are p olynomials P n ( z ) = z n + a n − 1 n z n − 1 + · · · + a 0 n , sat- isfying (2.1) Z Z D P n ( z ) P m ( z ) e − N V ( z ) d 2 z = h n δ mn for all m, n = 0 , 1 , 2 . . . . The follo wing lemma is standard. Lemma 2. Z N = N ! N − 1 Y n =0 h n . The pro of is exactly th e same as in th e case of the Hermitian mod el (see e.g. [1]). As in th e case of th e Hermitian mo del, Lemma 2 r educes the question of the asymptotic analysis of the partition fun ction Z N as N → ∞ to the asymptotic analysis of the orth ogonal p olynomials P n ( z ) as n, N → ∞ . 3. Ma trix ¯ ∂ -problem Using the orthogonal p olynomials on the line as an analogy (see [4], [1]), w e set (3.1) Y n ( z ) =    P n ( z ) 1 π R R D P n ( z ′ ) z ′ − z e − N V ( z ′ ) d 2 z ′ − π h n − 1 P n − 1 ( z ) − 1 h n − 1 R R D P n − 1 ( z ′ ) z ′ − z e − N V ( z ′ ) d 2 z ′    . It follo ws from the form ula ∂ ∂ ¯ z 1 z − z ′ = π δ ( z − z ′ ) , understo o d in the distributional sense, that (3.2) ∂ ∂ ¯ z Y n ( z ) = Y n ( z )( I − G ( z )) , OR THOGONAL POL YNOM IALS AND ¯ ∂ - PROBLEM 3 where I is 2 × 2 iden tit y matrix and (3.3) G ( z ) =  1 e − N V ( z ) χ D ( z ) 0 1  The follo wing prop osition is cen tral (cf. the case of the orthogonal p olyno- mials on the line). Prop osition 3. The matrix Y n ( z ) is the unique solution of the ¯ ∂ -pr oblem (3.2) – (3.3) with the normalization (3.4) Y n ( z ) =  I + O  1 z   z n 0 0 z − n  ≡  I + O  1 z  z nσ 3 , as | z | → ∞ , wher e σ 3 = ( 1 0 0 − 1 ) . Pr o of. It follo w s from the geometric series expansion 1 z − z ′ = 1 z ∞ X k =0  z ′ z  k as | z | → ∞ , and the prop erty (2.1 ), rewritten as (3.5) Z Z D P n ( z ) ¯ z m e − N V ( z ) d 2 z = h n δ mn , that the matrices (3.1) satisfy normalizat ion (3.4). Con ve rs ely , supp ose that the matrix Y ( z ) solves the ¯ ∂ -problem (3.2) with the asymptotics (3.4 ). It follo ws f r om the sp ecial form (3.3) of the matrix G ( z ) that ( Y n ) 11 ( z ) = P n ( z ) and ( Y n ) 21 ( z ) = Q n − 1 ( z ) — p olynomials of orders n and n − 1 resp ectiv ely , and ∂ ∂ ¯ z ( Y n ) 12 ( z ) = − P n ( z ) e − N V ( z ) χ D ( z ) , ∂ ∂ ¯ z ( Y n ) 22 ( z ) = − Q n − 1 ( z ) e − N V ( z ) χ D ( z ) . No w it follo ws from normalization (3.4) that the leading coefficient of p oly- nomial P n ( z ) is 1 and ( Y n ) 12 ( z ) = 1 π Z Z D P n ( z ′ ) z ′ − z e − N V ( z ′ ) d 2 z ′ , ( Y n ) 22 ( z ) = 1 π Z Z D Q n − 1 ( z ′ ) z ′ − z e − N V ( z ′ ) d 2 z ′ . 4 ALEXANDER R . ITS AND LEON A. T AKHT AJAN Using geometric series and normalization (3.4) once again, we ob tain that p olynomials P n and Q n − 1 satisfy Z Z D P n ( z ) ¯ z m e − N V ( z ) d 2 z = 0 for m < n, − Z Z D Q n − 1 ( z ) ¯ z m − 1 e − N V ( z ) d 2 z = π δ mn for m ≤ n. F rom here it follo ws that Z Z D P n ( z ) P m ( z ) e − N V ( z ) d 2 z = 0 for all m < n, whic h is sufficient to conclude that P n ( z ) are orthogonal p olynomials on C with the weigh t e − N V ( z ) χ D ( z ). Finally , p olynomials Q n ( z ) satisfy Z Z D Q n − 1 ( z ) P m − 1 ( z ) e − N V ( z ) d 2 z = − π δ mn for all m ≤ n, so that Q n ( z ) = − π h n P n ( z ).  Similar to the case of the usual orthogonal p olynomials, Prop osition 3 reduces th e asymp totic analysis of the orthogonal p olynomials (2.1) to th e asymptotic analysis of the solution of the ¯ ∂ -problem (3.2)-(3.4). 4. Elbau-Felder po tential. To w ards a normal ma trix version of the DKMVZ asy mptotic approa ch. W e will consider the matrix mo del with the w eigh t e − N V ( z ) χ D ( z ), where V ( z ) is the Elbau-F elder [3] p oten tial (4.1) V ( z ) = 1 t 0 | z | 2 − 2 Re n +1 X k =1 t k z k ! , where t 1 = 0, | t 2 | < 1 / 2 and t 0 V ( z ) is p ositiv e on D \ { 0 } . Using again the Hermitian matrix mo del analogy , we s hall exp ect th at a fun damen tal role in the asymp totic analysis of the ¯ ∂ -problem (3.2)-(3.4) will b e pla yed b y the e qu ilibrium me asur e . Definition 4. An equilibrium m easure for V on D is a Borel p r obabilit y measure µ on D without p oin t masses so that I ( µ ) = inf I ( ν ) , ν ⊂ M ( D ) , where M ( D ) is the set of all Borel pr obabilit y measures µ on D without p oint masses, and the functional I ( ν ) is defined b y the equation, I ( ν ) := Z V ( z ) dν ( z ) + Z Z z 6 = ζ log | z − ζ | − 1 dν ( z ) dν ( ζ ) . OR THOGONAL POL YNOM IALS AND ¯ ∂ - PROBLEM 5 Theorem 5 (Elbau-F elder [3]) . Ther e is δ > 0 such that for al l 0 < t 0 < δ the uni que e quilibrium me asur e dµ exi sts and is given by dµ ( z ) = 1 π t 0 χ D + ( z ) d 2 z , wher e the domain D + ⊂ D c ontains the origin and has the pr op e rty that t 0 = 1 π Z Z D + d 2 z , t k = − 1 π k Z Z C \ D + z − k d 2 z = 1 2 π ik I ∂ D + ¯ z z − k dz , k = 1 , . . . , n + 1 , t k = 0 , j > n + 1 . These r elations determines D + uniquely. In fact, the b oundary Γ of D + is a p olynom ial c urve of de gr e e n , i.e. Γ i s a smo oth simple close d curve in the c omplex plane with a p ar ametrization h : S 1 ⊂ C → C of the form, h ( w ) = r w + a 0 + a 1 w − 1 + ... + a n w − n , | w | = 1 , with r > 0 and a n 6 = 0 . The e quilib rium me asur e has the fol lowing pr op erties. Set E ( z ) = V ( z ) + 2 Z Z log | z − ζ | − 1 dµ ( ζ ) . 1. E ( z ) = E 0 — a c onstant — for z ∈ D + . 2. E ( z ) ≥ E 0 — for z ∈ D \ D + . 4.1. A naiv e DKMVZ sc heme. Sup p ose th at there is an analytic func- tion g ( z ) with the follo wing p rop erties. 1. g ( z ) = log z + O  1 z  as | z | → ∞ . 2. V ( z ) − g ( z ) − g ( z ) = E 0 on D + . 3. V ( z ) − g ( z ) − g ( z ) > E 0 on C \ D + . Suc h fu n ction g ( z ) could b e u sed to stud y the asymp totics of the matrix Y n ( z ) in the limit, (4.2) n, N → ∞ , n N = γ , γ is fixed , in exactly the same manner is it is done in the case of the orthogo nal p oly- nomials in the line (see [2] and [1]). Namely , set V γ ( z ) = 1 γ V ( z ) and consid er corresp onding equ ilibrium measure dµ γ (assuming that 0 < γ t 0 < δ ) w ith the domain D + ( γ ) and the corresp ondin g fu n ction g γ ( z ) satisfying prop erties 1–3 (with D + and E 0 replaced by D + ( γ ) and E 0 ( γ )). Then we can “undr ess” the ¯ ∂ -problem (3.2) –(3.3) with the n ormalization (3.4) b y setting Y n ( z ) = e − nE 0 ( γ ) 2 σ 3 Ψ n ( z ) e ng γ ( z ) σ 3 + nE 0 ( γ ) 2 σ 3 . 6 ALEXANDER R . ITS AND LEON A. T AKHT AJAN The r esulting matirx Ψ n ( z ) satisfies the simplified ¯ ∂ -problem (4.3) ∂ ∂ ¯ z Ψ n ( z ) = Ψ n ( z )  0 − e − n ( V γ ( z ) − g γ ( z ) − g γ ( z ) − E 0 ( γ )) 0 0  with the s tandard normalization (4.4) Ψ n ( z ) = I + O  1 z  as | z | → ∞ , whic h follo ws from prop ert y 1 of the function g γ ( z ). It is easy to pass to limit (4.2) in the ¯ ∂ -problem (4.3)–(4.4). Ind eed, it follo ws from prop er ties 2-3 of the function g γ ( z ) that lim n,N →∞ Ψ n ( z ) = Ψ 0 ( z ) , where the matrix Ψ 0 ( z ) satisfies the follo wing mo del ¯ ∂ -pr oblem . (4.5) ∂ ∂ ¯ z Ψ 0 ( z ) = Ψ 0 ( z ) ( ( 0 − 1 0 0 ) z ∈ D + ( γ ) 0 z / ∈ D + ( γ ) with the s tandard normalization (4.6) Ψ 0 ( z ) = I + O  1 z  as | z | → ∞ . This mod el ¯ ∂ -problem is easily solv ed explicitly , Ψ 0 ( z ) =   1 1 π R R D + ( γ ) 1 ζ − z d 2 ζ 0 1   . 4.2. F unction g ( z ) . Of course, the main assu mption that there is an an- alytic function function g ( z ) satisfying the prop erties 1-3 is not correct. Firstly , it follo ws from the pr op erty 1 that g ( z ) in th e neighb orh o o d of in- finit y is defined u p to an inte ger multiple of 2 π i , whic h is not a d ra wbac k since e ng ( z ) σ 3 si w ell-defined f or in teger n . Secondly , the prop ert y 2 implies that the fun ction V ( z ) is harmonic in D + , whic h clearly cont radicts (4.1). Adding to the confu sion is th e formal manipulation log | z − ζ | 2 = log( z − ζ ) + log( ¯ z − ¯ ζ ) , whic h suggests that (4.7) g ( z ) = Z Z log( z − ζ ) dµ ( ζ ) = 1 π t 0 Z Z D + log( z − ζ ) d 2 ζ satisfies prop erties 1-3 . Ho w ev er, this is not so since w e n eed to treat carefully the bran ches of log in order to defin e th e in tegral in (4.7 ) and in ve stigate its analytic p r op erties. OR THOGONAL POL YNOM IALS AND ¯ ∂ - PROBLEM 7 F or this aim, consider the logarithmic p oten tial giv en b y the uniform distribution of c harges in the domain D , V 0 ( z ) = Z Z D log | z − w | 2 d 2 w. Let Γ = ∂ D w ith fi xed p oint ζ 0 ∈ Γ. F or z ∈ D denote by D ε ( z ) d omain obtained b y remo ving the disk of radius ε around z , so that ∂ D ε ( z ) = Γ ∪ − C ε ( z ) , where C ε ( z ) is the circle | w − z | = ε orien ted counter-cl oskwise, and the min us sign denotes negativ e orienta tion. Sin ce log | w − z | 2 dw ∧ d ¯ w = − d (log | w − z | 2 ¯ wdw ) − ¯ w ¯ w − ¯ z dw ∧ d ¯ w, b y Stok es’ theorem, we hav e V 0 ( z ) = i 2 Z Z D log | z − w | 2 dw ∧ d ¯ w = i 2 lim ε → 0 Z Z D ε ( z )  − d (log | w − z | 2 ¯ wdw ) − ¯ w ¯ w − ¯ z dw ∧ d ¯ w  = 1 2 i lim ε → 0 Z Z D ε ( z ) d  log | w − z | 2 ¯ wdw + ¯ w ¯ w − ¯ z wd ¯ w  = 1 2 i lim ε → 0 I ∂ D ε ( z )  log | ζ − z | 2 ¯ ζ dζ + ¯ ζ ¯ ζ − ¯ z ζ d ¯ ζ  . No w lim ε → 0 Z C ε ( z ) log | ζ − z | 2 ¯ ζ dζ = 0 , lim ε → 0 Z C ε ( z ) ¯ ζ ¯ ζ − ¯ z ζ d ¯ ζ = − 2 π i | z | 2 , so that V 0 ( z ) = π | z | 2 + 1 2 i I Γ  log | ζ − z | 2 ¯ ζ dζ + ¯ ζ ¯ ζ − ¯ z ζ d ¯ ζ  . Set ω = ¯ w dw and define a function Ω on Γ \ { ζ 0 } by Ω( ζ ) = ζ Z ζ 0 ω , where the inte gration is along the oriented p ath in Γ connecting p oint s ζ 0 and ζ . W e ha ve Ω − ( ζ 0 ) = 0 for a p ath consisting of a s in gle p oin t ζ 0 , and Ω + ( ζ 0 ) = I Γ ¯ wdw = Z Z D d ¯ w ∧ dw = 2 iA ( D ) , 8 ALEXANDER R . ITS AND LEON A. T AKHT AJAN for the lo op Γ starting and ending at ζ 0 , wh ere A ( D ) is the area of D . Thus I Γ log | ζ − z | 2 ζ dζ = I Γ log | ζ − z | 2 d Ω( ζ ) = ∆(log | z − ζ | 2 Ω( ζ ))   ζ 0 ζ 0 − I Γ Ω( ζ )  dζ ζ − z + d ¯ ζ ¯ ζ − ¯ z  = 2 iA ( D ) log | z − ζ 0 | 2 − I Γ Ω( ζ )  dζ ζ − z + d ¯ ζ ¯ ζ − ¯ z  , so that V 0 ( z ) = π | z | 2 + A ( D ) log | z − ζ 0 | 2 + i 2 I Γ  Ω( ζ ) ζ − z dζ + Ω( ζ ) ¯ ζ − ¯ z d ¯ ζ − ¯ ζ ¯ ζ − ¯ z ζ d ¯ ζ  . Since the p oten tial V 0 is real-v alued, we h a v e V 0 ( z ) = π | z | 2 + A ( D ) log | z − ζ 0 | 2 + i 4 I Γ Ω( ζ ) ζ − z dζ − Ω( ζ ) ¯ ζ − ¯ z d ¯ ζ − Ω( ζ ) ζ − z dζ + Ω( ζ ) ¯ ζ − ¯ z d ¯ ζ + ζ ζ − z ¯ ζ dζ − ¯ ζ ¯ ζ − ¯ z ζ d ¯ ζ ! = π | z | 2 + A ( D ) log | z − ζ 0 | 2 + i 4 I Γ Ω( ζ ) − Ω( ζ ) + | ζ | 2 ζ − z dζ − Ω( ζ ) − Ω( ζ ) + | ζ | 2 ¯ ζ − ¯ z d ¯ ζ ! . Finally , observin g that ω + ¯ ω = d | w | 2 , w e get Ω( ζ ) + Ω( ζ ) = ζ Z ζ 0 d | w | 2 = | ζ | 2 − | ζ 0 | 2 , so that Ω( ζ ) − Ω( ζ ) + | ζ | 2 = 2Ω( ζ ) + | ζ 0 | 2 , and w e obtain (4.8) V 0 ( z ) = π ( | z | 2 − | ζ 0 | 2 ) + A ( D ) log | z − ζ 0 | 2 + i 2 I Γ Ω( ζ ) ζ − z dζ − Ω( ζ ) ¯ ζ − ¯ z d ¯ ζ ! , where z ∈ D . This is a desired representa tion of the area p oten tial V 0 ( z ) as the real p art of the first d eriv ative of a single la y er p otent ial. OR THOGONAL POL YNOM IALS AND ¯ ∂ - PROBLEM 9 The same computation for z ∈ C \ ¯ D gives (4.9) V 0 ( z ) = A ( D ) log | z − ζ 0 | 2 + i 2 I Γ Ω( ζ ) ζ − z dζ − Ω( ζ ) ¯ ζ − ¯ z d ¯ ζ ! , Returning to Elb au-F elder p otentia l V ( z ) and setting D = D + , Γ = ∂ D + , w e get E ( z ) = V ( z ) − 1 π t 0 V 0 ( z ), so that (4.10) V ( z ) − 1 t 0 ( | z | 2 − | ζ 0 | 2 ) − log | z − ζ 0 | 2 − i 2 π t 0 I Γ Ω( ζ ) ζ − z dζ − Ω( ζ ) ¯ ζ − ¯ z d ¯ ζ ! = E 0 when z ∈ D + , and (4.11) V ( z ) − log | z − ζ 0 | 2 − i 2 π t 0 I Γ Ω( ζ ) ζ − z dζ − Ω( ζ ) ¯ ζ − ¯ z d ¯ ζ ! = E ( z ) when z ∈ D − . R emark 6 . W e note that equation (4.10 ), i.e. the statemen t that the l.h.s. of (4.10) is constant wh en z ∈ D + , is equiv alen t to the moment equations of Theorem 5 wh ic h determine the con tour Γ (cf.[3], p.12, Lemma 6.3) . No w we are ready to in tro du ce the function g ( z ). Namely , set (4.12) g ( z ) = log( z − ζ 0 ) + i 2 π t 0 I Γ Ω( ζ ) ζ − z dζ . The function g ( z ) is holomorphic in C \ Γ, is multi- v alued w ith p erio d s 2 π i Z (singe-v alued on th e plane w ith the outside cut s tarting fr om ζ 0 ) and has the asymptotics g ( z ) = log z + O ( z − 1 ) as z → ∞ . The fun ction e ng ( z ) is single-v alued for n ∈ Z . The function g ( z ) is discon- tin uous on Γ (b y Sokhotski-Plemelj form ula). W e su mmarize this as the f ollo win g statement . Prop osition 7. The E lb au-F elder p otential V ( z ) has the fol lowing r epr e- sentations (i) F or z ∈ D + , V ( z ) − g ( z ) − g ( z ) = E 0 + 1 t 0 ( | z | 2 − | ζ 0 | 2 ) . (ii) F or z ∈ D − , V ( z ) − g ( z ) − g ( z ) = E ( z ) . (iii) F or z ∈ D \ D + , (4.13) V ( z ) − g ( z ) − g ( z ) = E ( z ) > E 0 . 10 ALEXANDER R . ITS AND LEON A. T AKHT AJAN 4.3. A first p ossible version of the DKMVZ sc heme. The correct strategy is now the follo wing. L et g γ ( z ), D + ( γ ), E 0 ( γ ), etc. denote the resp ectiv e ob jects asso ciated with the p oten tial V γ ( z ). W e set (4.14) Y n ( z ) = e − nE 0 ( γ ) 2 σ 3 + n | ζ 0 | 2 2 γ t 0 σ 3 Ψ n ( z ) e ng γ ( z ) σ 3 + nE 0 ( γ ) 2 σ 3 − n | ζ 0 | 2 2 γ t 0 σ 3 . The resulting m atirx Ψ n ( z ) satisfies the ¯ ∂ -problem (the correct version of (4.3)) ∂ ∂ ¯ z Ψ n ( z ) = Ψ n ( z ) 0 − e − n | z | 2 γ t 0 0 0 ! , z ∈ D + , (4.15) ∂ ∂ ¯ z Ψ n ( z ) = Ψ n ( z )   0 − e − n „ E ( z ) − E 0 ( γ )+ | ζ 0 | 2 γ t 0 « χ D + ( z ) 0 0   , z ∈ C \ D + , Ψ n + ( z ) = Ψ n − ( z ) e n γ t 0 Ω( z ) σ 3 , z ∈ Γ ≡ Γ( γ ) , with the s tandard normalization (4.16) Ψ n ( z ) = I + O  1 z  as | z | → ∞ . By virtue of condition (4.13 ), we exp ect that the limiting function Ψ 0 n ( z ) satisfies the mod el problem (4.17) ∂ ∂ ¯ z Ψ 0 n ( z ) = Ψ 0 n ( z )              0 − e − n | z | 2 γ t 0 0 0   z ∈ D + ( γ ) 0 z / ∈ D + ( γ ) Ψ 0 n + ( z ) = Ψ 0 n − ( z ) e n γ t 0 Ω( z ) σ 3 , z ∈ Γ , with the s tandard normalization (4.18) Ψ 0 ( z ) = I + O  1 z  as | z | → ∞ . The op en questions are no w the follo w ing. (1) Ho w to solv e this m o del p r oblem? Th e “un fortunate” thin g is the presence of complex conjugation in (4.17). Indeed, if we neglect the jump across the con tour Γ, the ¯ ∂ -problem alone can b e of course solv ed explicitly , (4.19) Ψ 0 ( z ) =    1 1 π R R D + ( γ ) e − n | ζ | 2 γ t 0 ζ − z d 2 ζ 0 1    , and the solution won’t h a v e an y jump s. If not for the complex conjugation, the fun ction Ψ 0 ( z ) could b e used to u ndress in the OR THOGONAL POL YNOM IALS AND ¯ ∂ - PROBLEM 11 usual w ay pr oblem (4.17) and reduce it to a pu r e Riemann-Hilb ert problem. (2) The arguments th at led us to the mo del p r oblem (4.17) and which are based on inequalit y (4.13 ), ev en on the form al lev el, are not v ery con vincing: the r eal part of Ω( z ) is | z | 2 − | ζ 0 | 2 6 = 0 so that the diagonal jump matrix on Γ is not pu re oscillatory . 4.4. A second p ossible version of the DKMVZ scheme. The ab o v e deficiency of the pr op osed analog of the DKMVZ sc heme can b e partially o v ercome by p erforming the follo wing mo difi cation. L et us rep lace the fun c- tion Ω( ζ ) by the function, Ω 0 ( ζ ) = 1 2 Z ζ ζ 0 ( ¯ wdw − w d ¯ w ) . The f unction Ω 0 ( ζ ) is pure imaginary on Γ and it is relate d to the function Ω( ζ ) b y the equation, Ω 0 ( ζ ) = Ω( ζ ) − | ζ | 2 − | ζ 0 | 2 2 . By using again Stok es’ theorem, w e observe that i 4 π t 0 I Γ ( | ζ | 2 − | ζ 0 | 2 )  dζ ζ − z − d ¯ ζ ¯ ζ − ¯ z  = − 1 2 π t 0 Z Z D +  ζ ζ − z + ¯ ζ ¯ ζ − ¯ z  d 2 ζ − 1 t 0 ( | z | 2 − | ζ 0 | 2 ) . This allo ws to re-write equations (4.10) and (4.1 1 ) in the form, V ( z ) − log | z − ζ 0 | 2 − i 2 π t 0 I Γ Ω 0 ( ζ ) ζ − z dζ − Ω 0 ( ζ ) ¯ ζ − ¯ z d ¯ ζ ! (4.20) + 1 2 π t 0 Z Z D +  ζ ζ − z + ¯ ζ ¯ ζ − ¯ z  d 2 ζ = E 0 when z ∈ D + , and V ( z ) − log | z − ζ 0 | 2 − i 2 π t 0 I Γ Ω 0 ( ζ ) ζ − z dζ − Ω 0 ( ζ ) ¯ ζ − ¯ z d ¯ ζ ! (4.21) + 1 2 π t 0 Z Z D +  ζ ζ − z + ¯ ζ ¯ ζ − ¯ z  d 2 ζ = E ( z ) when z ∈ D − . These formulae in turn yield the follo wing mo d ification of the definition (4.1 2 ) of the g -function (4.22) g ( z ) = log( z − ζ 0 ) + i 2 π t 0 I Γ Ω 0 ( ζ ) ζ − z dζ − 1 2 π t 0 Z Z D + ζ ζ − z d 2 ζ . 12 ALEXANDER R . ITS AND LEON A. T AKHT AJAN Note that th e function g ( z ) is not holomorphic in C \ Γ an ymore! In fact 1 , (4.23) ∂ ∂ ¯ z g ( z ) = z 2 t 0 χ D + ( z ) . A sligh t mo dification is also needed in the definition of the function Ψ n ( z ); indeed, w e should put, (4.24) Y n ( z ) = e − nE 0 ( γ ) 2 σ 3 Ψ n ( z ) e ng γ ( z ) σ 3 + nE 0 ( γ ) 2 σ 3 . T aking into accoun t (4.23), the ¯ ∂ -problem for the matrix Ψ no w reads, (4.25) ∂ ∂ ¯ z Ψ n ( z ) + n z 2 γ t 0 Ψ n ( z ) σ 3 = Ψ n ( z )  0 − 1 0 0  , z ∈ D + , (4.26) ∂ ∂ ¯ z Ψ n ( z ) = Ψ n ( z )  0 − e − n ( E ( z ) − E 0 ( γ )) χ D + ( z ) 0 0  , z ∈ C \ D + , Ψ n + ( z ) = Ψ n − ( z ) e n γ t 0 Ω 0 ( z ) σ 3 , z ∈ Γ ≡ Γ( γ ) , with the s tandard normalization (4.27) Ψ n ( z ) = I + O  1 z  as | z | → ∞ . The fun ction Ω 0 ( z ) is n o w purely imaginary . T herefore, the argu m en ts based on inequalit y (4.13) seem to b e more soun d than in the previous approac h, and they lead us to the follo w in g new mo del ¯ ∂ -problem (4.28) ∂ ∂ ¯ z Ψ 0 n ( z ) + n z 2 γ t 0 Ψ 0 n ( z ) σ 3 = Ψ 0 n ( z )  0 − 1 0 0  , z ∈ D + , (4.29) ∂ ∂ ¯ z Ψ 0 n ( z ) = 0 , z ∈ C \ D + , Ψ 0 n + ( z ) = Ψ 0 n − ( z ) e n γ t 0 Ω 0 ( z ) σ 3 , z ∈ Γ ≡ Γ( γ ) , with the s tandard normalization (4.30) Ψ 0 n ( z ) = I + O  1 z  as | z | → ∞ . R emark 8 . Due to the p resence of the large parameter n in the left h and side of equation (4.25), the transition to the mo del problem (4.2 8 )-(4.30) is still not qu ite satisfacto ry ev en on th e formal lev el. 1 Probably this prop erty of th e function g ( z ) reflects a ma jor difference b etw een th e Riemann-Hilb ert problem and th e ¯ ∂ -problem. OR THOGONAL POL YNOM IALS AND ¯ ∂ - PROBLEM 13 4.5. An imp ortant concluding rema rk. In con text of the theory of or- thogonal p olynomials, a matrix ¯ ∂ -problem has also app eared in the recen t w ork of K . McLaughlin and P . Miller [6] dev oted to orthogonal p olynomials on the unite circle with the non-analytic weigh ts. Ho w ev er, unlik e the p rob- lem (3.2)-(3.4), the ¯ ∂ -problem of [6] is not the starting p oin t of the analysis; indeed, the staring p oin t of [6] is still the usual matrix Riemann-Hilb ert problem an d the ¯ ∂ -problem of McLaughlin and Miller is introduced out of the necessit y to mo dify th e “op enning lenses” step of the usual DKMVZ sc heme. Even more imp ortan t difference b et w een the ¯ ∂ -problem consid ered here and the ¯ ∂ -problem in [6] is the absence of the complex conjugation in the basic ¯ ∂ -relation. In one hand, this fact simplifies the implement ation of the “undressin g pro cedures” — the v ery imp ortant tec hn ical elemen t of all in tegrable asymptotic schemes. On the other h and, as w e ha ve s h o wn, the presence of the complex conjugation in the righ t h and side of (3.2) is truly essen tial for the incorp oration in to the asym p totic analysis of the concepts of equilibrium measure and g -fu n ction. In spite of th ese differences we b eliev e that using metho ds of [6] will help allo w to o v ercome the indicated ab o v e obstacles in the asymptotic analysis of the ¯ ∂ -problem (3.2)-(3.4). Sp ecifically , w e w e th in k that one needs to dev elop and then apply to th e mo del pr oblems (4.17)-(4.18) or (4.28)-(4.30) of the the ¯ ∂ -v ersion of the “op enn in g lenses” s tep in th e DKMVZ method . In conclusion, w e wan t to p oint out at the ¯ ∂ -metho d in the theory of in tegrable systems, introd uced long ago b y A.S. F ok as and M.J. Ablo witz in [5], as yet another sour ce of tools for th e analysis of the ¯ ∂ -problem (3.2 )- (3.4). A cknowledgement s Alexander I ts w as supp orted in part by the NSF grant s DMS-040 1009 and DMS-0701 768, and L eon T akht a jan — by the NSF gran ts DMS-02 04628 and DMS-0705 263. Referen ces [1] P . A. Deif t, Orthogonal P olynomials and Rand om Matrices: A Riemann- Hilb ert A p- proac h, Cour ant L e ctur e Notes i n Mathematics , 3 , CIMS, New Y ork (1999). [2] P . Deift, T. Kriecherbauer, K. T-R. McLaughlin, S . V enakides, and X. Zh ou, Uniform asymptotics for p olynomials orthogonal with resp ect to v arying exp onential wei ghts and applications to u niversal ity questions in random matrix th eory , Commun. Pur e Appl. Math. , 52 ( 1999), 1335-1425. [3] P . Elbau a nd G. F elder, Density of eigen v alues of ra nd om normal matrices, Commun. Math. Phys , 259 (2005), 433-450. [4] A.S. F oka s, A.R . Its and A .V . Kitaev, The isomono dromy approach to matrix prob- lems in 2D q u antum gravit y , Commun. Math. Phys. , 147 (1992), 395–430 . [5] A.S. F ok as and M.J. Ab lo witz, On the inv erse scattering transform of multidimen- sional nonlinear equations related to first-order systems in the plane, J. Math. Phys. , 25 (1984), 2494-2505. 14 ALEXANDER R . ITS AND LEON A. T AKHT AJAN [6] K. T.-R. McLaughlin and P . D. Miller, The ¯ ∂ steep est descen t method and t he as- ymptotic b ehavior of polynomials orth ogonal on the unit circle with fixed and exp o- nentiall y v arying nonanalytic weig hts, IMPR , v. 2006 (2006), 1-78. [7] P . B. Wiegmann, A . Zabrod in , Conformal maps and integrable hierarc hise, Commun. Math. Phys. , 213 ( 2000), 523-538. [8] I. K . Kosto v, I. Krichever, M. Mineev-W einstein, P . B. Wiegmann, A . Zabrodin, τ - function for analytic cu rves, R andom matrix mo dels and their appli c ations , 285-299, Math. Sci. R es. Inst. Publ., 40 , Cambridge Univ. Press, Cambridge, 2001. Dep ar tment of Ma the ma tical S ciences, Indiana University-Purdue Univer- sity Indiana polis, Indi anapolis, IN 46202-3216, USA E-mail addr ess : itsa@math.iupui .edu Dep ar tment of Ma thema tics, S tony Brook University, Stony Broo k, NY 11794-365 1, US A E-mail addr ess : leontak@math.su nysb.edu