Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution

Geometry of the Ising p ersistence problem and the univ ersal Bonnet-Manin P ainlev ´ e VI distribution Iv an Dornic 1,2* and Rob ert Con te 3,4 1 Service de ph ysique de l’ ´ etat condens ´ e (UMR 3680), Univ ersit ´ e P aris-Saclay , CEA Sacla y , CNRS, Gif-sur-Yv ette, 91191, F rance. 2 Lab oratoire de ph ysique th ´ eorique de la mati ` ere condens ´ ee (UMR 7600), Sorb onne univ ersit ´ e, CNRS, 4, Place Jussieu, P aris, 75252 Paris Cedex 05, F rance. 3 Cen tre Borelli, LR C MESO, Universit ´ e P aris-Saclay , ENS Paris-Sacla y , CNRS, 4, av en ue des sciences, Gif-sur-Yv ette, 91190, F rance. 4 Departmen t of Mathematics, The Universit y of Hong Kong, P okfulam, Hong Kong, China. *Corresp onding author(s). E-mail(s): Iv an.Dornic@cea.fr, OR CID h ttps://orcid.org/0000-0001-9036-8893 ; Con tributing authors: Rob ert.Con te@cea.fr, OR CID h ttps://orcid.org/0000-0002-1840-5095 ; Abstract W e determine the full p ersistence probability distribution for a non-Marko vian sto c hastic pro cess, motiv ated by ﬁrst-passage questions arising in in teracting spin systems and allied systems. W e sho w that this distribution is gov erned by a dis- tinguished Painlev ´ e VI system arising from an exact F redholm Pfaﬃan structure asso ciated with the integrable se ch kernel, K sech = 1 / (2 π cosh[( x − y ) / 2]) . The universal p ersistence exp onen t originally obtained by Derrida, Hakim and P asquier is reco v ered as an asymptotic observ able and acquires a natural geomet- ric interpretation. In the stationary scaling regime, the p ersistence probability admits an exact Pfaﬃan decomp osition into even and o dd F redholm determi- nan ts of the in tegrable se ch kernel. These determinan ts are con trolled b y a unique global solution of a second-order nonlinear ordinary diﬀerential equation, which is identiﬁed as a particular P ainlev´ e VI equation. The corresponding Painlev ´ e VI connection problem determines the p ersistence exp onen t as a limiting v alue at inﬁnit y . 1 W e further show that the P ainlev´ e VI system go verning p ersistence admits a direct geometric interpretation: the relev ant solution coincides with the mean curv ature of a one-parameter family of Bonnet surfaces immersed in R 3 . A folding transformation b et ween such surfaces singles out the Painlev ´ e VI equation with Manin co eﬃcien ts [0 , 0 , 0 , 0] , which in particular go verns the univ ersal p ersis- tence distribution in the symmetric Ising case. In this framew ork, the p ersistence exp onen t is iden tiﬁed with the asymptotic mean curv ature of the associated surface. De dic ate d to Jo el L. L eb owitz, the gr e at soul of statistic al physics Accepted for publication in J. Stat. Phys. (2026) ᾿ Αγεωμέτρητ ος μηδεὶς εἰσί τ ω ( “L et no one ignor ant of ge ometry enter her e.” ) [Legendary [ 102 ] inscription written at the en trance of Aristotle’s classroom in Plato’s Academ y .] Keyw ords Keyw ords: Persistence probability; ﬁrst-passage pro cesses; F redholm determinants and Pfaﬃans; integrable k ernels; P ainlev ´ e VI equation; Bonnet surfaces 02.30.Ik – Integrable systems 33.e17 – Painlev ´ e-t yp e functions 02.50.Cw – Probability theory 02.40.Hw Classical diﬀerential geometry Con ten ts 1 In tro duction 3 1.1 Statemen t of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Relation to existing literature . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Outline of the pap er . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 In tegrable structure of p ersistence probability: pro of strategy 10 2.1 F redholm or Pfaﬃans determinan ts and gap probabilities . . . . . . . 10 2.2 In tegrable structure and closed ODE system ob ey ed b y the resolv ent k ernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 In tegration in terms of a system of intert wined P VI functions . . . . . 18 2.4 The distinguished Painlev ´ e VI go verning p ersistence . . . . . . . . . . 20 2.5 T ranscenden tal nature of the p ersistence distribution . . . . . . . . . . 21 3 Conclusion 21 2 A The DHP formula for the p ersistence probability as a Pfaﬃan gap- spacing probability with the sech kernel 23 B The Painlev ´ e VI solutions for the mean curv ature and the metric of Bonnet surfaces in R 3 24 C Three remark able second order nonlinear ODEs: P VI , H VI , C VI 40 D The Boro din-Okounk ov formula and the solution of the Bonnet P VI connection problem 43 1 In tro duction What is the chance for a ﬂuctuating quantit y to hav e always remained ab o ve its long- term tendency or, conv ersely , the likelihoo d it ﬁrst crosses its av erage v alue at a given time? The study of the ﬁrst-pass age prop erties for a random pro cess revolv es around such questions, with inn umerable applications in the natural sciences [ 9 , 86 , 99 ]. In the con- text of interacting man y-bo dy nonequilibrium systems, in particular those displaying coarsening dynamics [ 20 ], this topic has attracted a lot of interest in the 1990’s under the name of “p ersistence” (see [ 1 , 21 , 79 ] for reviews, the last tw o recen t ones). Probably one of the main reasons for the upsurge of activity for this sub ject, and the fascination for it, was sparked b y the disco v ery that even for the simplest p ossible mo dels [ 40 – 43 , 78 ], the p ersistence probability decays algebraically for large times, with an exp onen t, generally denoted θ , which does not seem to b e related to an y other kno wn static or dynamic critical exp onen ts, although it subsumes in a simple num b er the dynamics of the in terwo v en mosaic of gro wing domains. In 1995-1996, Derrida, Hakim, and Pasquier (DHP) obtained in [ 41 , 42 ] an ex- traordinary exact analytical expression for a con tin uous family b θ ( q ) of persistence exp onen ts in a protot ypical model of nonequilibrium statistical ph ysics, the q -state P otts mo del in one space dimension evolving with zero-temp erature Glaub er dynam- ics. If p 0 ([ t 1 , t 2 ]; q ) is the probability that the Potts spin at the origin of a semi-inﬁnite c hain has nev er ﬂipp ed b et ween times t 1 and t 2 , they sho wed that when 1 ≪ t 1 ≪ t 2 this probabilit y deca ys as p 0 ([ t 1 , t 2 ]; q ) ∝ ( t 1 /t 2 ) b θ ( q ) / 2 , where b θ ( q ) = − 1 8 + 2 π 2  arccos  2 − q √ 2 q  2 , (1.1) with the striking limiting v alue b θ (2) / 2 = 3 / 16 in the q = 2 Ising case. Muc h later, in 2018, Popla vskyi and Schehr [ 97 ] obtained directly this particular p ersistence exp onent 3 / 16, and revealed it was universal, since it app ears in several a priori unrelated problems, such as the determination of the real zeros of the Kac p olynomial, the ﬁrst-passage properties for a D = 2 randomly diﬀusing ﬁeld, or within the truncated orthogonal ensemble of random matrix theory . In the present work, we determine the full probability law giving for the D = 1 Ising-P otts model and allied pro cesses their univ ersal persistence distribution as a 3 distinguished sixth Painlev ´ e function, recov ering in particular the exp onent ( 1.1 ) as an asymptotic observ able. Using instead the equiv alen t but more conv enient viewp oin t of ± Ising spins on a D = 1 chain ev olving from an arbitrary initial magnetization − 1 ≤ m ≤ 1, we sho w that in the appropriate stationary scaling regime the p ersistence probability distribution function,  7→ P 0 (  ; m ), is go verned by a one-parameter family of genuinely transcenden tal solutions H = H ( x ; ξ ) for a particular sixth P ainlev ´ e equation (P VI ), P 0 (  ; m ) = Det  Id − ξ K + sech  ↾ [0 ,ℓ ] = exp Z ℓ 0 d x H ( x ; ξ ) − p −H ′ ( x ; ξ ) 2 ! , (1.2) with ξ = ξ ( m ) = 1 − m 2 , and H ′ = d H / d x . The structure of this F redholm determinant evidences that the p ersistence prob- abilit y is a Pfaﬃan gap-spacing probability generating function, here asso ciated to a translation-in v ariant and in tegrable scalar k ernel, the se ch kernel, K sech ( x 1 − x 2 ) : = 1 2 π 1 cosh [( x 1 − x 2 ) / 2] , x 1 − x 2 ∈ R . (1.3) In the formula ( 1.2 ), (Id − ξ K + sech ) ↾ [0 ,ℓ ] is the integral op erator for the sech kernel restricted to its even eigenfunctions on the interv al [0 ,  ]. F or Ising spins, the generating function parameter b ecomes a thinning parameter, ξ = ξ ( m ) ∈ (0 , 1), which keeps trac k of the a verage magnetization in the random initial condition. Our analysis further establishes the existence of a unique negativ e and decreasing solution x 7→ H ( x ; ξ ), regular at the origin, deﬁned on the p ositive real axis and globally b ounded for each ﬁxed parameter ξ = ξ ( m ) ≤ 1. Its limiting v alue at inﬁnity , κ ( m ) : = − lim x → + ∞ H ( x ; ξ ( m ) ) = − 1 8 + 2 π 2 " arccos r m 2 2 !# 2 , (1.4) is entirely determined b y the asso ciated P ainlev´ e VI c onne ction pr oblem for H , and therefore depends only on m 2 . It gov erns the exponential deca y P 0 (  ; m ) ∝ e − κ ( m ) ℓ/ 2 of the p ersistence probabilit y on the stationary timescale  ≫ 1. The dep endence on the sign of the magnetization does not enter in to the de- ca y rate κ ( m ) itself, but is entirely enco ded at the probabilistic lev el b y a Pfaﬃan parit y decomp osition, which constitutes the ﬁrst main result of this work and mixes probabilistic and analytic ingredients. In particular, the Painlev ´ e VI connection problem b ecomes singular at ξ = 1, where the F redholm determinants for the sech kernel dev elop a Fisher-Hartwig singularity [ 37 , 49 ], and this leads in the symmetric Ising case m = 0 to the remark able ratio- nal v alue κ (0) / 2 = 3 / 16. The Derrida–Hakim–Pasquier exp onen t ( 1.1 ) is reco vered asymptotically from ( 1.4 ) by com bining the analytic determination of the Painlev ´ e VI connection constan t κ ( ξ ) with the probabilistic identiﬁcation 1 /q = (1 + m ) / 2 relating P otts and Ising initial conditions on the p ositive branch. But which Painlev ´ e VI e quation? 4 The determination of the precise Painlev ´ e VI system gov erning the p ersistence problem is a cen tral issue of the presen t w ork, and ultimately justiﬁes the title of the article. Bey ond the mere fact that the function H en tering the F redholm representa- tion ( 1.2 ) satisﬁes a closed second-order, second-degree nonlinear ordinary diﬀerential equation that we shall obtain b y standard T racy-Widom tec hniques for in tegrable op erators, one must determine which Painlev ´ e VI is actually in volv ed. This amoun ts to identifying the v alues of the four parameters α, β , γ , δ , the bound- ary conditions selecting the relev an t solution, and the nature — classical or genuinely transcenden tal — of the solution required for the p ersistence problem. A stunning outcome of the present w ork is that the function H determining the p ersistence probabilit y admits a direct geometric interpretation. More precisely , H can b e identiﬁed with the mean curv ature of a remark able family of surfaces in three– dimensional Euclidean space discov ered b y Bonnet in 1867. Within this framew ork, the asso ciated Painlev ´ e VI equation arises as the Gauss– Co dazzi compatibility condition of the moving frame, while geometry reveals non trivial transformation properties of the p ersistence system. In particular, it provides a natural explanation for the app earance of an an- gle arccos in the Derrida–Hakim–Pasquier exp onen t ( 1.1 ), evocative of a nonlinear analogue of Buﬀon’s needle problem. More imp ortan tly , the geometric viewp oin t leads to a folding transformation b e- t ween distinct Bonnet Painlev ´ e VI surfaces, which mak es it p ossible to express the required function H = H ( x ; m ) in terms of a function Q = Q ( t ) solution of ¨ Q = 1 2  1 Q + 1 Q − 1 + 1 Q − t  ˙ Q 2 −  1 t + 1 Q − 1 + 1 Q − t  ˙ Q, (1.5) (with ˙ Q = d Q/ d t ) corresp onding to a P VI equation having the simplest possible parameters [ α, β , γ , δ ] = [0 , 0 , 0 , 0] . (1.6) A Painlev ´ e VI equation with the v ery same co eﬃcients w as encountered by Manin in 1995 in a completely diﬀeren t algebro–geometric con text. It thus app ears fair to us to christen the particular solution for m = 0 giving the full p ersistence distribution function P ⋆ 0 (  ) = P 0 (  ; m = 0) with its universal deca y rate κ ⋆ : = κ (0) / 2 = 3 / 16 as the Bonnet–Manin Painlev´ e VI , after the names of these tw o great geometers from the past centuries. W e no w turn to a precise mathematical statemen t of the results contained in this w ork. W e presen t them as tw o main theorems: the ﬁrst combines probability and analysis, the second analysis and geometry , with the ov erarching Painlev ´ e VI structure pro viding the bridge betw een them. 1.1 Statemen t of results Theorem 1.1 (Pfaﬃan decomposition and F redholm representation) . In the sta- tionary sc aling r e gime and starting fr om m -magnetize d r andom initial c onditions, 5 the p ersistenc e pr ob ability P + 0 (  ; m ) that the Ising spin lo c ate d at the origin of semi- inﬁnite chain evolving with zer o-temp er atur e Glaub er dynamics has always b e en in the + state for a length of time  admits, along with its twin pr ob ability P − 0 (  ; m ) , an exact Pfaﬃan p arity de c omp osition P + 0 (  ; m ) = P − 0 (  ; − m ) = 1 + m 2 D + + D − 2 + 1 − m 2 D + − D − 2 . (1.7) Her e, D ± = D ± (  ; ξ ) , with ξ = ξ ( m ) = 1 − m 2 , ar e the F r e dholm determinants for the even and o dd p arts of the thinne d se ch kernel r estricte d to [0 ,  ] : D ± (  ; ξ ) : = Det  Id − ξ K ± sech  ↾ [0 ,ℓ ] = exp Z ℓ 0 d x H ( x ; ξ ) ∓ p −H ′ ( x ; ξ ) 2 ! . (1.8) Both ar e determine d by the unique solution H ( x ) = H ( x ; ξ ) r e gular at the origin, ne gative, de cr e asing, and glob al ly b ounde d on R + for the Cauchy pr oblem         H ′′ 2 H ′ + coth x  2 + 1 (sinh x ) 2 H 2 H ′ + 2(coth x ) H + H ′ = 1 4 , x > 0 , H (0 + ) = − ξ and H ′ (0 + ) = −H 2 (0 + ) , x ↘ 0 , 0 < ξ ≤ 1 . (1.9) In the sc aling r e gime, the p ersistenc e distribution P + 0 (  ; m ) for this Ising spin is r elate d to p 0 ([ t 1 , t 2 ]; q ) , the one for a Potts spin, thr ough the identiﬁc ation P + 0 (  ; m ) = 1 q p 0 ([ t 1 , t 2 ]; q ) ,  = log ( t 2 /t 1 ) > 0 , 1 q = 1 + m 2 , (1.10) valid for arbitr ary  > 0 and m ≥ 0 , which implies that κ ( m ) = − lim x →∞ H ( x ; ξ ( m )) , the ne gative limiting value at inﬁnity attaine d by the solution to the ODE ( 1.9 ), gives b ack the half-sp ac e DHP p ersistenc e exp onent: P + 0 (  ; m ) ∝ e − κ ( m ) ℓ/ 2 ,  ≫ 1 = ⇒ p 0 ([ t 1 , t 2 ]; q ) ∝ ( t 1 /t 2 ) b θ ( q ) / 2 , t 1 /t 2 ≪ 1 . (1.11) W e recall that, as ﬁrst emphasized by Derrida, Hakim and Pasquier in [ 42 ], the p ersistence problem must be formulated on a semi-inﬁnite chain in order for the un- derlying Pfaﬃan structure to emerge: only in this geometry do the tw o half-lines on eac h side of a spin which nev er ﬂips evolv e indep enden tly , leading to a factorization of the p ersistence probabilit y on the inﬁnite chain and to a p ersistence exponent whic h is t wice that obtained in the semi-inﬁnite setting. As for the equiv alence b et ween q -state Potts spins and ± Ising ones on a c hain ev olving with zero-temperature Glauber dynamics from random initial conditions, it is a w ell-kno wn probabilistic fact, see, e.g. [ 75 ]. Indeed, by declaring that one Potts color, 6 o ccurring with probabilit y 1 /q , corresp onds to an Ising spin +, while the remaining q − 1 colors group ed together corresp ond to an Ising spin − , one obtains an Ising c hain with magnetization m such that 1 /q = (1 + m ) / 2. In the following, w e shall exclusiv ely use the Ising language for the persistence problem, notably b ecause this corresp ondence provides a natural extension to non-integer v alues of q = 2 / (1 + m ). Equation ( 1.7 ) makes transparent how the Pfaﬃan structure propagates the Bernoulli weigh ts (1 ± m ) / 2 of the initial condition through the stationary scaling regime, by deforming them in to a diagonal superp osition of the even and o dd F redholm determinan ts. Of course, this formula is equiv alent to P ± 0 = D + ± m D − , showing that the dynamics eﬀectiv ely separates the t wo parit y sectors. Summing o ver the t w o p ossi- ble states a spin can stay without ﬂipping, one recov ers for the p ersistence probability distribution the expression ( 1.2 ) as a single F redholm Pfaﬃan determinant, P 0 (  ; m ) = P + 0 (  ; m ) + P − 0 (  ; m ) = D + (  ; ξ ) , ξ = 1 − m 2 , (1.12) namely the one for the ev en part of the sec h k ernel. Theorem 1.2 (Painlev ´ e VI structure of p ersistence and global solution of Bonnet surfaces) . (i) Ge ometric interpr etation. Ther e exists a one-p ar ameter family of immerse d surfac es Σ ξ ⊂ R 3 admitting c onformal c o or dinates ( z , ¯ z ) such that the func- tion H = H ( x ; ξ ) app e aring in The or em 1.1 c oincides with the me an curvatur e of Σ ξ . The surfac es Σ ξ b elong to the class of Bonnet surfac es, for which b oth the me an cur- vatur e function and the metric dep end on the sole r e al variable x = ℜ z . This yields ther efor e a glob al solution of the Gauss–Co dazzi e quations for Σ ξ . (ii) Painlev´ e VI avatars and tr ansformations. The quantities asso ciate d with the p ersistenc e pr ob ability ( H , 1 2 ( H ∓ √ −H ′ ) , and appr opriate c ombinations of the lo g- arithmic derivatives of D ± ) satisfy intertwine d Painlev´ e VI e quations. These e quations ar e bir ational ly or algebr aic al ly r elate d, p ossess c ontiguous mono dr omy exp onents, and admit a natur al ge ometric interpr etation on Σ ξ , as summarize d in T able 1 . (iii) F olding tr ansformation and Manin p ar ameters. A folding tr ansformation r elating distinct Bonnet surfac es singles out a distinguishe d Painlev´ e VI e quation with p ar ameters [ α, β , γ , δ ] = [0 , 0 , 0 , 0] , pr eviously identiﬁe d by Manin. This p articular Painlev ´ e VI e quation governs the universal p ersistenc e distribution P ⋆ 0 (  ) = P 0 (  ; 0) . (iv) Asymptotic curvatur e and p ersistenc e exp onent. F or al l ξ = ξ ( m ) , the p ersistenc e exp onent κ ( m ) is e qual to minus the asymptotic me an curvatur e of Σ ξ at inﬁnity. This limit c orr esp onds to the unique umbilic p oint of Σ ξ , wher e the two princip al curvatur es b e c ome e qual. This last sen tence is the reason why the persistence exponent is here denoted κ (like a geometric curv ature) and not θ as usual. Another reason is the standard notation θ for the mono dromy exp onents of P VI whic h determine the four parameters of the 7 generic P VI equation, ( θ 2 α , θ 2 β , θ 2 γ , θ 2 δ ) : = (2 α, − 2 β , 2 γ , 1 − 2 δ ) , (1.13) a con ven tion which resp ects the parit y under any p erm utation of the lo cations j = ∞ , 0 , 1 , t of the four deﬁning singularities of P VI . This implies three imp ortan t features for the present universal probability la w. 1. It is not related by an y birational or algebraic transformations to the “usual” Picard case [ 95 ] with all its monodromy exp onents { θ j } equal to zero (hence with co eﬃcien ts [ α, β , γ , δ ] = [0 , 0 , 0 , 1 / 2]), and whose general (tw o-parameter initial conditions dependent) solution is kno wn explicitly; 2. It is transcendental in the 19-th century sense (nonalgebraic dep endence in the tw o constan ts of in tegration); 3. It is not reducible to classical h ypergeometric solutions of P VI already encountered in a random matrix con text [ 51 ]. T able 1 Corresp ondence b et ween p ersistence, sixth Painlev ´ e function, and the t wo principal curv atures κ 1 , κ 2 of the Bonnet surface. The successive columns display: F redholm determinan ts for the sech kernel as deﬁned in ( 1.8 ), signed monodromy exponents { θ α , θ β , θ γ , θ δ } for each asso ciated P VI , geometric quantit y . F redholm determinan t Mono drom y exp onen ts Geometric quan tity D ± / D ∓ { 0 , 0 , 0 , 1 } K = κ 1 κ 2 (total curv ature) D + × D −  1 2 , 0 , 0 , 1 2  H = κ 1 + κ 2 2 (mean curv ature) D ±  1 4 , 1 4 , 1 4 , 1 4  L = κ 1 − κ 2 2 (skew curvature) 1.2 Relation to existing literature The persistence problem can be view ed as a ﬁrst-passage question for a non-Mark ovian sto c hastic pro cess, a p erspective going back to early studies of stationary Gaussian pro cesses. In the context of in teracting man y-b o dy systems, this viewp oin t gained prominence in the 1990s through coarsening dynamics, where p ersistence quan tiﬁes the probability that a single degree of freedom never changes its state as macroscopic domains form and evolv e. The in trinsically non-Mark ovian nature of this problem w as quic kly recognized as a ma jor obstacle to exact calculations, making the result obtained b y Derrida, Hakim and Pasquier a genuine tour de force: their exact expression for the p ersistence exp onen t in the one-dimensional Potts mo del remained for a long time an isolated ac hievemen t. More than t wen ty y ears later, a decisive adv ance w as made b y P opla vskyi and Sc hehr [ 97 ], who computed directly the Ising p ersistence exp onen t 3 / 16 and demon- strated its universalit y b y sho wing that it appears in sev eral a priori unrelated problems. These systems are Gaussian (at least asymptotically) and share the same 8 non-Mark ovian stationary correlator C ( T 2 − T 1 ) : = E [ X ( T 1 ) X ( T 2 ) ] = sec h( T 2 − T 1 ) . (1.14) Their work ﬁrmly established the univ ersal nature of the exp onen t, but did not address the full p ersistence distribution nor its analytic structure. Rigorous asymptotic results for F redholm Pfaﬃans asso ciated with integrable ker- nels w ere later obtained b y FitzGerald, T rib e and Zaboronski [ 50 ]. Their analysis pro vides a precise mathematical framework for large-gap asymptotics and clariﬁes the role of Fisher–Hartwig singularities in Pfaﬃan p oin t pro cesses, but do es not iden tify the nonlinear diﬀerential equations gov erning the corresponding probabilit y la ws. Finally , in an unpublished man uscript [ 45 ] motiv ated b y the results of [ 97 ], the ﬁrst author observ ed that the Ising p ersistence probabilit y admits a F redholm determinant represen tation in volving the in tegrable sec h kernel. How ever, neither the solution of the asso ciated P ainlev´ e VI connection problem nor the full underlying geometric structure in terms of intert wined P VI w ere identiﬁed at that stage. The k ey adv ance of the presen t w ork is to recognize that the persistence probabilit y can be iden tiﬁed with a standard ob ject of random matrix theory: a conditional gap- spacing probabilit y endo wed with a Pfaﬃan structure for the integrable kernel K sech , th us essentially the sec h correlator ( 1.14 ). This identiﬁcation provides direct access to the full and universal persistence distribution and to its gov erning P ainlev´ e VI system, since for the particular non-Mark ovian Gaussian stationary pro cess with the correlator ( 1.14 ), w e pro ve that its conditional ﬁrst-passage probability distribution function is P  inf 0 0     X (0) = 0  = Det  Id − K + sech  ↾ [0 ,ℓ ] ≡ P ⋆ 0 (  ) . (1.15) 1.3 Outline of the pap er The pap er is organized so as to progressively unfold the in tegrable structure underlying p ersistence. W e ﬁrst establish a probabilistic and analytic formulation of the p ersistence prob- lem in terms of F redholm Pfaﬃans and a closed nonlinear diﬀerential system. This leads to the pro of of Theorem A, which isolates the Pfaﬃan parity decomp osition and the associated second-order second-degree nonlinear ordinarry diﬀeren tial equation (ODE) satisﬁed by the logarithmic deriv ativ es of the F redholm Pfaﬃans determinants. W e then reinterpret this system geometrically , iden tifying the relev an t Painlev ´ e VI functions with the mean curv ature of Bonnet surfaces in R 3 . This viewpoint culminates in Theorem B, whic h clariﬁes the rˆ ole of folding transformations, mono drom y data, and the distinguished Manin parameters [ α, β , γ , δ ] = [0 , 0 , 0 , 0]. Sev eral tec hnical complements, asymptotic analyses, and comparisons with existing results are gathered in the remaining sections and appendices. 9 2 In tegrable structure of p ersistence probabilit y: pro of strategy The pro ofs of Theorems 1.1 and 1.2 rely on a combination of probabilistic, analytic, and geometric argumen ts. Rather than presen ting them in a purely axiomatic manner, w e hav e organized the presentation according to the underlying integrable structure, progressing from general prop erties of Pfaﬃan p oin t processes to the speciﬁc features of the p ersistence k ernel. The argumen t proceeds through the follo wing steps: • In Section 2.1 , we recall the basic properties of determinantal and Pfaﬃan point pro cesses, coming from from a family of probabilit y con volution k ernels deﬁned on the real line or the unit circle, and which naturally arise in the p ersistence problem. • In Section 2.2 , we consider the in tegrable kernel ( 2.38 ) with a general parame- ter θ and derive a closed diﬀeren tial system gov erning the asso ciated F redholm determinan ts. • Section 2.3 is devoted to the explicit in tegration of this system and to its form ulation in terms of Painlev ´ e functions. • In Section 2.4 , w e sp ecialize to the v alue θ = 1 / 2 relev ant for pers istence and sho w ho w the resulting system can b e reduced, via a folding symmetry , to the particular P ainlev´ e VI equation iden tiﬁed b y Manin. • Finally , Section 2.5 establishes the genuinely transcendental nature of the resulting univ ersal distribution. 2.1 F redholm or Pfaﬃans determinan ts and gap probabilities Ev ery member of the family of translation-inv ariant even kernel functions of the generic form ( 2.38 ) deﬁnes a point pro cess either on the unit circle (when ν 2 < 0) or on the real line (when ν 2 > 0). In the latter case, note that ν is thus an arbitrary real scaling factor, while in the former one it is of course constrained b y 2 π . F or concrete applications, one has typically ν = i or ν = 1 or ν = 2 with the resp ectiv e kernels of reference K N or K θ : both choices hav e pros and cons [ 113 , p 66], and w e shall use b oth, depending whether one tak es the righ tmost endp oin t of the symmetric in terv al [ − T , T ] as the indep enden t v ariable, or its length  = 2 T . Since, the ODEs ob ey ed b y the as sociated resolv ents contain terms coth(2 ν T ) ≡ coth( ν  ), it should b e clear whic h choice is in use from the corresponding equations. W e shall everywhere follow the terminology of T racy-Widom [ 115 ] (section I I): “If K ( x, y ) is the k ernel of an integral op erator K then we shall sp eak interc hangeably of the determinan t for K ( x, y ) or K , and the determinant of the operator Id − K .” In order to simplify the notation in the presen t section, we simply denote K ( x ) an y k ernel K θ ( ν x ). Eac h of this con tin uous, bounded, and symmetric k ernel when it acts on some compact interv al I = [ T 1 , T 2 ] deﬁnes a linear op erator K ↾ [ T 1 ,T 2 ] on L 2 ( I , d x ), which we also write as K ↾ I , or ev en just as K ↾ . This op erator is alwa ys lo cally trace-class for an y ﬁnite interv al length, and its eigen v alues are simple, discrete, 10 non-negativ e, b ounded, and ordered as 1 > λ 0 ( I ) > λ 1 ( I ) > · · · > 0 . (2.16) The F redholm determinant generating function is obtained b y multiplying the k ernel by ξ , D ([ T 1 , T 2 ]; ξ ) = ∞ X n =0 ( − ξ ) n n ! Z T 2 T 1 d x 1 · · · Z T 2 T 1 d x n det 1 ≤ j,k ≤ n [ K ( x j − x k ) ] (2.17) and it is equal to the sp ectral determinan t D ([ T 1 , T 2 ]; ξ ) = Det ( Id − ξ K ) ↾ [ T 1 ,T 2 ] = Y n ( 1 − ξ λ n ( I ) ) . (2.18) By translation in v ariance of the kernel, it suﬃces to consider this determinan t on the symmetric in terv al [ − T , T ] of length  = 2 T around the origin, in which case the ev en and odd parts of the kernel K ± ( x, y ) : = 1 2 ( K ( x, y ) ± K ( − x, y ) ) ≡ 1 2 ( K ( x − y ) ± K ( x + y ) ) , (2.19) select the even and o dd square-in tegrable eigenfunctions: Z T − T d y K ± ( x, y ) f n ( y ) = Z T 0 d y ( K ( x − y ) ± K ( x + y )) f n ( y ) = λ n ( T ) f n ( x ) , n even / odd . (2.20) In terms of the resolv ent op erator R for ξ K ↾ , deﬁned as usual through Id + R = (Id − ξ K ↾ ) − 1 , (2.21) the corresponding F redholm determinan ts are [ 115 , Eq. (30)] D ± ([ − T , T ]; ξ ) : = Det  Id − ξ K ±  ↾ [ − T ,T ] = exp − Z T 0 d x [ R ( x, x ; ξ ) ± R ( − x, x ; ξ ) ] ! . (2.22) Of course, the pro duct of these tw o, for short D = D + .D − , gives for a symmetric in terv al the F redholm determinan t ( 2.18 ) for ξ K ↾ , whic h is th us such that D ([ − T , T ]; ξ ) = Y n even and o dd (1 − ξ λ n ( T )) = exp − 2 Z T 0 d x R ( x, x ; ξ ) ! . (2.23) F or any ξ K ↾ [ − T ,T ] op erator on a symmetric interv al with an ev en k ernel func- tion K ( x − y ) = K ( y − x ), the resolven t kernel functions at coincident and opp osite 11 endp oin ts obey the symmetry relations R ( x, x ; ξ ) = R ( − x, − x ; ξ ) , R ( x, − x ; ξ ) = R ( − x, x ; ξ ) , − T < x < T , (2.24) along with a crucial diﬀerential constrain t which w as ﬁrst found by Gaudin (but nev er published!) for the sine kernel, but which remains v alid [ 85 , Appendix A16] for an y ev en-diﬀerence kernel function d d x R ( x, x ; ξ ) = 2 [ R ( − x, x ; ξ ) ] 2 , − T < x < T . (2.25) It is therefore suﬃcien t to consider these tw o resolven t functions for 0 < x < T , and even more conv enien tly [ 113 , p.65] to view them as functions of the (v ariable) righ tmost endp oin t T of the interv al. Therefore we abbreviate and deﬁne for all T > 0 R ( T ) : = lim x → T − R ( x, x ; ξ ) , R (0 + ) = ξ K (0) , (2.26) the second equalit y coming from the Neumann expansion of the resolven t ( 2.21 ), while from Gaudin’s relation ( 2.24 ) ev aluated for x → T − , w e set S ( T ) : = lim x → T − R ( − x, x ; ξ ) = ⇒ S ( T ) = p R ′ ( T ) / 2 , S (0 + ) = R (0 + ) (2.27) after choosing the + sign when solving backw ards for S in ( 2.24 ). F or both these functions, the thinning parameter is now hidden in their v alue at the origin T = 0, along with the normalization chosen for the density ρ 1 = K (0) (see ( 2.29 ) b elo w) of the stationary p oint pro cess generated by the translation-inv ariant kernel. All the determinan ts D, D + , and D − can thus b e reconstructed after quadratures from the sole knowledge of the function S ( T ), S ( T ) = 1 2 r − d 2 d T 2 log D ([ − T , T ]; ξ ) . (2.28) Finally , we recall the deﬁnitions of the “inclusive” and “exclusive” correlation functions. The ﬁrst are giv en b y ρ n ( x 1 , x 2 , . . . , x n ) = det 1 ≤ j,k ≤ n [ K ( x j − x k ) ] , (2.29) so that the v alue for ξ = 1 of the F redholm determinan t ( 2.17 ) when expanded, namely D ([ T 1 , T 2 ]; 1) = 1 − Z T 2 T 1 dx 1 ρ 1 + 1 2! Z T 2 T 1 Z T 2 T 1 dx 1 dx 2 ρ 2 ( x 1 , x 2 ) − · · · , (2.30) is the gap-sp acing pr ob ability , i.e. the probability that the interv al [ T 1 , T 2 ] is void of an y points for this stationary point pro cess with (constan t) av erage densit y ρ 1 = K (0). 12 As for the second (or J´ anossy) correlation functions, they are deﬁned by ω n ( x 1 , x 2 , . . . , x n ; ξ ) = D ([ T 1 , T 2 ]; ξ ) . det 1 ≤ j,k ≤ n [ R ( x j , x k ; ξ ) ] . (2.31) In particular, these exclusive correlation functions yields for the (unthinned) p oin t pro cess generated by the k ernel on I = [ T 1 , T 2 ] the probability P [ exactly m points in I ] = Det ( Id − K ↾ I ) 1 m ! Z T 2 T 1 d x 1 · · · Z T 2 T 1 d x m det 1 ≤ j,k ≤ n [ R ( x j , x k ) ] . (2.32) After division b y the F redholm determinan t for K on [0 , T ], whic h from ( 2.23 ) is e − R T 0 R (note the absence of factor 2 this time), the case m = 1 in ( 2.32 ) sho ws that ω 1 ( T )d T : = R ( T )d T is the c onditional probability to ﬁnd the ﬁrst point in ( T , T + d T ) (and none on [0 , T ]), while the case m = 2 is the conditional probability density to ﬁnd a p oin t at 0, another one at T , and none in b et ween. Using the symmetries ( 2.24 ), this join t conditional densit y function is therefore ω 2 ( T ) = R 2 ( T ) − S 2 ( T ) , (2.33) a relation which turns out to giv e the square of the lo cal sk ew curv ature L for our Bonnet-P VI surfaces, cf. T able 1 , hence their global Willmore energy up on in tegration. 2.2 In tegrable structure and closed ODE system ob ey ed b y the resolven t k ernels Tw o kno wn k ernels in random matrix theory hav e giv en rise to F redholm determinan ts and gap-spacing probabilit y expressed by some P VI function. The ﬁrst one is the famous circular unitary ensemble CUE N in tro duced b y Dyson [ 46 ], K N ( x 1 − x 2 ) : = 1 2 π sin [ N ( x 1 − x 2 ) / 2 ] sin ( x 1 − x 2 ) / 2 , N integer = 1 , 2 , . . . , − π < x 1 − x 2 ≤ π , (2.34) whic h gives the eigenv alues of a random unitary matrix U ( N ) distributed according to the Haar measure. The second one has been considered b y Nishigaki [ 90 ] to study some “critical” random matrix ensembles: K ( x 1 − x 2 ) = a sin [ π ( x 1 − x 2 ) ] π sinh [ a ( x 1 − x 2 ) ] , x 1 − x 2 ∈ R . (2.35) F or the sech kernel of central in terest for persistence, since 1 cosh ( x/ 2) = 2 sinh ( x/ 2) 2 sinh ( x/ 2) cosh ( x/ 2) = 2 sinh ( x/ 2) sinh ( x ) , (2.36) 13 a natural observ ation is the identit y in the complex plane provided one w aives the constrain t N integer, K sech ( x 1 − x 2 ) = K N (2i( x 1 − x 2 ))    N =1 / 2 . (2.37) W e therefore consider a common extrap olation to these tw o k ernels acting either on the real line or on the unit circle, also containing the sec h kernel. Such an extrapolation is the following family of translation-inv arian t ev en k ernels, K θ ( ν ( x 1 − x 2 )) : = ρ θ sinh [ θ ν ( x 1 − x 2 ) ] θ sinh [ ν ( x 1 − x 2 ) ] , ρ θ > 0 , θ 2 , ν 2 ∈ R . (2.38) In this section, we establish the system of ODEs asso ciated to this k ernel. Let us ﬁrst explain the roles of the t wo parameters ν 2 and θ . According to the sign of ν 2 , the determinantal p oin t pro cess deﬁned by this kernel is either on the circle ( ν 2 < 0) or on the real line ( ν 2 ≥ 0). In b oth cases, the normalization factor ρ θ is determined by the condition that K θ alw ays b e an even probabilit y distribution function: 1 = R K θ . As to the real parameter θ , it will app ear to be the unique mono drom y parameter c haracterizing the P ainlev´ e P VI o ccurring for Bonnet surfaces. Note that, when the kernel acts on the real line ( ν 2 > 0), the parameter ν can be scaled out. Tw o c hoices we shall employ are ν = 1 or ν = 2 for reasons whic h will be made clear below. In all cases, one m ust restrict − 1 < θ < 1 to ensure the probabilit y normalization R R K θ = 1: K θ ( x 1 − x 2 ) : = cot ( π θ / 2) π sinh [ θ ( x 1 − x 2 )] sinh ( x 1 − x 2 ) , − 1 < θ < 1 , x 1 − x 2 ∈ R , (2.39) It is also worth noticing the w ell-deﬁned point wise limit θ → 0 for the latter kernel, K 0 ( x 1 − x 2 ) : = 2 π 2 x 1 − x 2 sinh [( x 1 − x 2 )] = lim θ → 0 K θ ( x 1 − x 2 ) , (2.40) whic h exists thanks to the probability normalization, and whic h will b e sho wn to yield the Bonnet-Manin P VI . All these kernels are the simplest mem b ers of a class of exp onential variants of in tegrable op erators K ↾ I in tro duced and studied by T racy and Widom [ 113 ]. These op erators hav e k ernels of the form K ( x, y ) χ I ( y ) , K ( x, y ) = φ ( x ) ψ ( y ) − φ ( y ) ψ ( x ) e 2 ν x − e 2 ν y , (2.41) in which χ is the c haracteristic function of some in terv al I = [ T 1 , T 2 ] (or of a union of disjoin t in terv als), and the v ector ( φ, ψ ) t is the general solution for z ∈ C of the 2 × 2 14 matrix ﬁrst-order ODE d d z  φ ψ  = M  φ ψ  , M =  a b c d  . (2.42) If one deﬁnes for x ∈ I the t wo auxiliary functions of one v ariable [ 64 , Eq. (5.5)],  Φ( x ; [ T 1 , T 2 ]) = ⟨ x | (Id − K ) − 1 | φ ⟩ , Ψ( x ; [ T 1 , T 2 ]) = ⟨ x | (Id − K ) − 1 | ψ ⟩ , (2.43) using the conv enien t bra-ket notation, then the scalar resolven t R ( x, y ), i.e. the kernel of (Id − K ) − 1 K ↾ I , can b e written lik e the scalar kernel K ( x, y ), ∀ x ∈ [ − T , T ] , ∀ y ∈ [ − T , T ] , R ( x, y ) = Φ( x )Ψ( y ) − Φ( y )Ψ( x ) e 2 ν x − e 2 ν y · (2.44) Moreo ver, if the elements ( a, b, c, d )( z ) of M ob ey a linearizable ODE system, then, as prov en by Its, Izergin, Korepin, Sla vnov [ 64 ] and T racy and Widom [ 113 ], there exists a closed system of nonlinear in tegrable partial diﬀerential equations (PDE), whose dependent v ariables are scalar pro ducts built from the resolv ent, and whose indep enden t v ariables are the end p oin ts of the in terv al(s). This PDE system is the generalization for this class of integrable k ernels of the one found for the sine kernel b y JMMS [ 65 ]. In our case, the family of probability-con volution kernels ( 2.38 ) is generated simply b y taking a constan t matrix M conjugated to one with eigen v alues ν (1 ± θ ) P − 1 M P = ν  1 + θ ∗ 0 1 − θ  , (2.45) where w e k eep a Jordan form with a non-zero upper-right element so as to hav e alw ays t wo linearly indep enden t solutions φ, ψ even in the degenerate case θ → 0. Compared to the previous section ( 2.1 ), w e simplify the notation for the resolv ent k ernel R ( x, y ; ξ ) to R ( x, y ), and w e contin ue to denote b y R ( T ) and S ( T ) the resp ectiv e limits as x → T − of the resolven t kernel at coinciden t and opposite endpoints: the thinning parameter ξ is tak en in to accoun t b y the v alue at the origin T = 0 of these t wo functions R and S , cf. ( 2.26 ), ( 2.27 ), and (of course !) it does not app ear explicitly in the ODEs they ob ey we are going to establish: since the in terv al I = [ T 1 , T 2 ] = [ − T , T ] depends on only one v ariable, the PDE system degenerates to an ODE system, in tegrable by construction. This ODE system is six-dimensional, and its dep enden t v ariables are the six ﬁelds R ( T ), S ( T ) deﬁned through ( 2.27 ), and the four scalar pro ducts p j ( T ) , q j ( T ), j = 1 , 2 whic h are obtained b y taking in ( 2.43 ) the limits x → T j , with T 1 = − T , T 2 = T ,  p 1 ( T ) = ⟨− T | (Id − K ) − 1 | φ ⟩ , p 2 ( T ) = ⟨ T | (Id − K ) − 1 | φ ⟩ , q 1 ( T ) = ⟨− T | (Id − K ) − 1 | ψ ⟩ , q 2 ( T ) = ⟨ T | (Id − K ) − 1 | ψ ⟩ , (2.46) 15 whic h ob ey a closed diﬀeren tial system [ 64 , Eqs. (8.16)–(8.17)]. First, the limits ( x, y ) → ( T , − T ) , ( T , T ) , ( − T , − T ) in ( 2.44 ) generate three algebraic (nondiﬀeren tial) equations      ( x → + T , y → − T ) : S ( T ) = p 1 q 2 − p 2 q 1 e 2 ν T − e − 2 ν T , ( x → + T , y → + T ) : 2 ν e +2 ν T R ( T ) = polynomial( p j , q j , p ′ j , q ′ j ) , ( x → − T , y → − T ) : 2 ν e − 2 ν T R ( T ) = polynomial( p j , q j , p ′ j , q ′ j ) , (2.47) with p j ( T ) , q j ( T ), j = 1 , 2 deﬁned in ( 2.46 ). Secondly , the deriv atives of p j and q j      d q 1 d T = − aq 1 − bp 1 + 2 S q 2 , d p 1 d T = − cq 1 − dp 1 + 2 S p 2 , d q 2 d T = aq 2 + bp 2 + 2 S q 1 , d p 2 d T = cq 2 + dp 2 + 2 S p 1 , (2.48) do not introduce any new function in the system. T o summarize, the closed diﬀeren tial system is made of the four ﬁrst order ODEs ( 2.48 ) for p j ( T ) , q j ( T ) and three algebraic equations b et w een ( p j , q j , R, S ),    p 1 q 2 − p 2 q 1 − 2 sinh(2 ν T ) S = 0 , b ( p 2 1 + p 2 2 ) − c ( q 2 1 + q 2 2 ) + ( a − d )( p 1 q 1 + p 2 q 2 ) − 4 ν cosh(2 ν T ) R − 4 sinh(2 ν T ) S 2 = 0 , b ( p 2 2 − p 2 1 ) − c ( q 2 2 − q 2 1 ) + ( a − d )( p 2 q 2 − p 1 q 1 ) − 4 ν sinh(2 ν T ) R = 0 . (2.49) By deriv ation mo dulo ( 2.48 ) of the three equations ( 2.49 ) algebraic in p j , q j , R , S , one generates three more equations algebraic in p j , q j , R, S, R ′ , S ′ (with ′ = d / d T ). The further algebraic elimination of the four deriv ativ es p ′ 1 , q ′ 1 , p ′ 2 , q ′ 2 and of the four ﬁelds p 1 , q 1 , p 2 , q 2 generates t wo ODEs only in volving R and S , whose co eﬃcien ts, as exp ected, just inv olv e the co eﬃcien ts of the matrix M through its constant sp ectral in v ariants tr M and det M . The ﬁrst of these ODEs is the one of Gaudin ( 2.50 ) R ′ − 2 S 2 = 0 , (2.50) whic h in this framework is reco vered as a c onse quenc e of the ab o v e diﬀerential system. Indeed, the sole condition tr M = a + d = ν coming from the speciﬁc form ( 2.45 ) enforces that the kernel is an ev en-diﬀerence one K ( x, y ) = K ( x − y ) = K ( y − x ). The second one can be nicely written as the sum of four squares, [ sinh(2 ν T ) R ] ′ 2 − [ sinh(2 ν T ) S ] ′ 2 − [ 2 ν sinh(2 ν T ) R ] 2 + [ 2 ν θ sinh(2 ν T ) S ] 2 = 0 . (2.51) Let us no w establish the ODEs ob ey ed b y the three quan tities R , S and R ± S , whic h, according to ( 2.22 ), respectively determine the v alues of the three determinan ts in T able 1 . It turns out that the equation for R ± S is independent of the sign chosen: the distinction is hidden in the initial conditions. These three equations, esp ecially for 16 the last tw o of them, are more compactly display ed using three intermediate functions using the temp orary notations ρ R = σ S = µ R ± S = sinh(2 ν T ) , (2.52) so that in particular ( 2.51 ) acquires the very simple form (where ′ = d / d T ) ρ ′ 2 − σ ′ 2 − (2 ν ρ ) 2 + (2 ν θ σ ) 2 = 0 . (2.53) W e remark that the common sinh factor in the deﬁnitions abov e is essentially the mo dulus of the Hopf factor in geometry , cf. App endix B . W e hav e also to emphasize that these notations are used only in this subsection, and that this ρ = ρ ( T ) b ears no relation to the inclusive n -point correlation functions ρ n deﬁned in Eq. ( 2.29 ) of the previous subsection, or that σ = σ ( T ) is distinct from the so-called Ok amoto–Jimbo- Miw a sigma-form Eq. ( C3 ) satisﬁed b y the reduced P VI Hamiltonian. The second order ODE for R = R ( T ), the resolven t kernel function at coincident p oin ts, is still manageable without the sinh factor: ( R ′′ + 4 ν coth(2 ν T ) R ′ ) 2 − 4 R ′ h 2 R ′ 2 + 8 ν coth(2 ν T ) RR ′ + 4 ν 2 θ 2 R ′ − 8 ν 2 (1 − coth 2 (2 ν T )) R 2 i = 0 , (2.54) One of the adv an tage to display it this w ay is that up to a simple rescaling it coincides direcly with the equation for the mean curv ature of Bonnet surfaces. Another one is that for θ = N and ν = i, where the coth functions ab o ve become ordinary trigono- metric ones, it is iden tical to [ 113 , Eq. (5.70)], which provides a useful chec k of the algebra. The ODE for µ ( T ) is indep enden t of the sign in R ± S : ( µ ′′ − 2 ν coth (2 ν T ) µ ′ ) 2 − 4 µ ′ 3 sinh (2 ν T ) +  4 ν 2 (1 − θ 2 − coth 2 (2 ν T )) + 8 2 ν coth (2 ν T ) sinh (2 ν T ) µ  µ ′ 2 + 8 ν 2 sinh (2 ν T )  2 ν θ 2 cosh (2 ν T ) − 2 µ  µµ ′ − 16 ν 4 θ 2 µ 2 = 0 . (2.55) Finally , the second order ODE for S ( T ) is ( σ ′′ + 8 σ 3 − (2 ν ) 2 θ 2 σ ) 2 − 16 coth 2 (2 ν T ) σ 2 h σ ′ 2 + 4 σ 4 − (2 ν ) 2 θ 2 σ 2 i = 0 . (2.56) As to the thinning parameter ξ , evidently absent of these three ODEs, it is determined b y the Neumann expansion of the resolven t. 17 2.3 In tegration in terms of a system of in tertwined P VI functions The in tegration of thes e second order second degree ODEs is straightforw ard. The metho d is to ﬁrst lo ok at the mov able poles of R , S and R ± S , then at the second order ODE deﬁned by the ﬁrst square of the Gauss decomposition of their ODE. Given these t wo essential pieces of information, one selects in the existing classiﬁcations of second order second degree ODEs whic h ha ve the Painlev ´ e property [ 23 , 26 , 35 , 36 ] the v ery few possible matc hes. Finally , one tries to iden tify the ODE at hand and the selected matc h b y a homographic transformation of the dep enden t v ariable, see details in [ 34 ]. The ODE for S has only four mo v able simple p oles with residues ± 1 / 2 and ± (i / 2) / sinh((2 ν ) T ) and its ﬁrst square deﬁnes an elliptic ODE with tw o simple p oles, therefore it is likely to b e equiv alent to a particular equation C VI of Chazy [ 26 , p 342], whose general solution w ( x ) is recalled in Appendix C . The ODE for R (resp. R ± S ) has only one mov able simple p ole of residue − 1 / 2 (resp. − 1) and its ﬁrst square deﬁnes a linear ODE, therefore it is likely to be equiv alent to a particular so-called sigma-form of P VI [ 26 , p 340], ( C3 ) whose general solution h ( s ), recalled in App endix C , is deﬁned so that h ( s ) / ( s ( s − 1)) is a tau-function of P VI (one mo v able simple pole of residue unit y). Both guesses are true, and the explicit integration is the follo wing. W e start with the most fundamental ob ject, namely S , b ecause, although it does not describe a probabilit y la w, it is essentially identical to the most intrinsic ob ject in geometry , namely the metric function G . Moreov er, given S , the probability la ws R and R ± S follo w by p erforming one quadrature of the Gaudin relation ( 2.50 ), cf. ( 2.28 ). Up to scaling factors, the equations ( 2.54 ), ( 2.56 ) ob ey ed by the re solv en t func- tions R ( T ) and S ( T ) are identical to ( B20 ), ( B21 ), resp ectiv ely satisﬁed b y the mean curv ature and the metric function of Bonnet surfaces, see App endix B . Up to rescaling, the ODE ( 2.56 ) is identical to ( w ′′ − 2 w 3 − D 2 (2 ν ) 2 w − D 3 (2 ν ) 3 ) 2 +  2 coth(2 ν T )  w − D 1 (2 ν ) cosh(2 ν T )  2 × h w ′ 2 − w 4 − D 2 (2 ν ) 2 w 2 − 2 D 3 (2 ν ) 3 w − D 4 (2 ν ) 4 i = 0 , D 1 = D 3 = D 4 = 0 , S = 2 iw sinh(2 ν T ) , (2.57) an equation ﬁrst isolated b y Chazy [ 26 , Eq. (C,V) p. 342] (follo wing the con ven tion of Chazy , the factor i ensures that w ( T ) admits t w o simples poles of residues ± 1). The mono drom y exp onents of the asso ciated P VI tak e t w o sets of v alues (with θ α = θ β ) (see App endix C ),     1 4 , 1 4 , θ 2 , 0  or  ( θ − 1) 2 4 , ( θ − 1) 2 4 , θ 2 4 , θ 2 4  = ( θ 2 α , θ 2 β , θ 2 γ , θ 2 δ ) or ( θ 2 α , θ 2 β , θ 2 δ , θ 2 γ ) . (2.58) 18 F or R , the map betw een ( 2.54 ) and ( C3 ) is R ( T ) = 2 ν h R ( s ) , s = 1 + coth(2 ν T ) 2 , (2.59) and the monodromy exp onen ts of the corresponding P VI are an yone of the t wo equiv alent sets ( θ 2 α , θ 2 β , θ 2 γ , ( θ δ − 1) 2 ) = (0 , θ 2 , θ 2 , 0) or ( θ 2 , 0 , 0 , θ 2 ) . (2.60) Both R and R ± S are Hamiltonians ev aluated on the equations of motion of tw o P VI with diﬀeren t monodromy quadruplets and distinct independent v ariables,      H VI ( p ( s ) , q ( s ) , s )d s = 2 R d T = h R d s s (1 − s ) , s or 1 − s = 1 + coth(2 ν T ) 2 , (2.61) and      H VI ( P ( t ) , Q ( t ) , t )d t = ( R ± S )d T = d t t (1 − t )  h ± + t 16 − θ 2 8  , t or 1 t = coth 2 ( ν T ) , (2.62) in whic h the subscripts remind that the tw o h ’s are diﬀerent. The ODE for h ± ( t ) is the particular case { θ α , θ β , θ γ , θ δ } = (1 / 2)( θ , θ , 1 − θ , 1 − θ ), see Eq. ( C3 ) in Appendix C . F or reference, the precise correspondence betw een the tw o resolv ent functions and the geometric quantities is −H ( x ) = 2 R ( x, x ; ξ ) = 2 ν h ( s ( x )) , s ( x ) = 1 + coth ( ν x ) 2 (2.63) G ( x ) = sinh ( ν x ) ν 2 R ( − x, x ; ξ ) = G ( t ( x )) , t ( x ) = tanh 2 ( ν x/ 2) (2.64) R emark . The p ersistence exp onent κ ( m ) is also c haracterized by an intrinsic ge- ometric inv ariant, namely the Willmore energy of the corresp onding Bonnet surface, deﬁned as the integral of H 2 − K o ver the whole surface, E ( m ) : = Z Σ > d A  H 2 − K  = 2 π H (Re z ; 1 − m 2 )    Re z =0 + Re z =+ ∞ = 2 π κ ( m ) − 1 − m 2 4 . (2.65) R emark . The inv ariance of ( 2.57 ) under parit y only requires D 1 = D 3 = 0. The consideration of the metric of a Bonnet surface in the Riemannian manifold R 3 ( c ) [ 13 , page 120] instead of the ﬂat Euclidean space R 3 allo ws b oth D 2 and D 4 to b e nonzero, the coeﬃcient D 4 b eing prop ortional to the extrinsic curv ature. 19 2.4 The distinguished P ainlev´ e VI go verning p ersistence The k ernel of p ersistence corresp onds to θ = 1 / 2. In this case (and only in this case), one of the admissible c hoices for the signed quadruplet { θ α , θ β , θ γ , θ δ } in ( 2.58 ) is { θ α , θ β , θ γ , θ δ } =  1 4 , 1 4 , 1 4 , 1 4  . (2.66) (W e recall that signs of the monodromy exp onents do matter in the Hamiltonian form ulation of P VI , in particular for the associated impulsion, see App endix C .) Then there exists an algebraic transformation leaving P VI form-in v ariant (i.e. only c hanging its four parameters), due to Kitaev [ 73 ], recalled as Prop osition C.1 in App endix C , whose t wo successive applications map this quadruplet to { 0 , 0 , 0 , 1 } ,  1 4 , 1 4 , 1 4 , 1 4   →  1 2 , 0 , 0 , 1 2   → { 0 , 0 , 0 , 1 } . (2.67) This folding transformation has later b een sho wn by Manin [ 81 ] to be iden tical to a Landen transformation b et ween elliptic functions in the elliptic representation of P VI . The ab ov e quadruplet ( 2.67 ) displa ys t wo adv antages, due to the cancellation of the last three terms of the impulsion p Eq. ( C10 ). In tro ducing the Lagrangian asso ciated to the Hamiltonian ( C8 ), L VI ( ˙ q , q , t ) : = p ˙ q − H VI ( p ( t ) , q ( t ) , t ) . (2.68) the ﬁrst adv antage is a m ultiplication b y a factor four of the tw o resp ectiv e Lagrangians, a prop erty symbolically written as 4 L { 4 × 1 / 4 } ( ˙ q , q , t ) = L { 3 × 0 , 1 } ( ˙ Q, Q, t ) (2.69) where the P VI function Q = Q ( t ) app earing on the right-hand-side abov e has there- fore Manin’s P VI co eﬃcien ts [ α, β , γ , δ ] = [ 0 , 0 , 0 , 0 ] . The second adv an tage is the exceptional equalit y of the Lagrangian and the Hamiltonian for the quadruplet of mono drom y exponents { 0 , 0 , 0 , 1 } . This justiﬁes what w e hav e announced m uch earlier in our Introduction and in our Theorem 1.2 : the o ccurrence of Manin’s P VI with these distinguished coeﬃcients for the p ersistence problem. Notice ﬁnally that since one of the admissible set of the mono drom y exp onen ts for the mean curv ature of a K θ -Bonnet surface can b e tak en as (cf. Prop osition B.1 in App endix B ) { θ α , θ β , θ γ , θ δ } = { θ , 0 , 0 , 1 − θ } , this Manin P VI is also a Bonnet P VI : the one with θ = 0 determined by the kernel K 0 , Eq. ( 2.40 ), whose F ourier transform by ( D7 ) is itself essentially the square of the (self-dual) sec h-kernel K sech ! 20 2.5 T ranscendental nature of the p ersistence distribution Due to the presence of these very special symmetries, one could w onder if the Bonnet-Manin P VI con trolling the p ersistence distribution function is a genuinely tran- scenden tal one, or a (particularly) conv oluted classical one, namely whether one could express it through some ﬁrst order or linearizable ODE, or as an algebraic function. Indeed, since there exists a c hoice of signs of the four mono dromy exp onen ts making their sum unity , the solution of the asso ciated P VI could b e the logarithmic deriv ative of a h yp ergeometric function. It could also b e one of the 48 algebraic solutions of P VI [ 76 ]. W e shall giv e tw o diﬀerent justiﬁcations to sho w this is not the case, thereby establishing the transcenden tal nature of the p ersistence probability densit y . The crux of the matter for b oth arguments is that S , the second logarithmic of the F redholm determinan t for the sec h kernel, cannot v anish. First pr o of . According to its link ( C14 ) betw een S and C VI , the impulsion P of the Hamil- tonian H VI Eq. ( C8 ) is nonzero, therefore the hypergeometric p ossibilit y (which is deﬁned b y P = 0) is ruled out. There remain to discard p ossible algebraic solutions. According to the list of [ 76 ], the only algebraic solution matching the mono drom y exp onen ts (1 / 4 , 1 / 4 , 1 / 4 , 1 / 4) is Q = √ t . This would again imply a zero impulsion and therefore S = 0, yielding a contradiction. □ Se c ond pr o of . The second pro of relies on a result of Borodin and Ok ounko v (App endix D ). Giv en any hyperb olic kernel K θ in the class ( 2.38 ), and thus in particular for the sech kernel there exist scalars A, B , C such that ∀ ξ , 0 ≤ ξ < 1 ∀ T : log Det(Id − ξ K sech ) ↾ [ − T ,T ] = − 2 Z T 0 d xR ( x ) = A T + B + C ( T ) , (2.70) in which C ( T ) is the F redholm determinan t of a k ernel whic h v anishes exp onentially fast as T → + ∞ . T aking the second deriv ativ e w.r.t. T , one obtains ∀ T : d 2 d T 2 log Det(Id − ξ K sech ) = 2 R ′ ( T ) = − 4 S 2 ( T ) = d 2 d T 2 C ( T ) , (2.71) and this second deriv ative of C ( T ) and S ( T ) are never equal to zero, cf. ( D25 ). □ 3 Conclusion In this work, w e ha ve sho wn that the full p ersistence probabilit y distribution for the one-dimensional Ising–Potts mo del in the stationary scaling regime is gov erned b y a distinguished P ainlev´ e VI system, whic h w e hav e termed the Bonnet–Manin Painlev ´ e VI . F rom a tec hnical standpoint, this result places the p ersistence problem within the general framew ork of integrable F redholm Pfaﬃans. In this resp ect, the se ch kernel determines the Bonnet–Manin Painlev ´ e VI in a wa y which is structurally similar to the wa y that the sine kernel determines the Gaudin–Mehta Painlev ´ e V distribution in random matrix theory . A t a more conceptual lev el, our results suggest that the p ersistence distribu- tion o ccupies, for nonequilibrium coarsening dynamics, a role analogous to that of 21 T racy–Widom distributions in equilibrium and near-equilibrium statistical systems: a universal la w characterized b y an in tegrable kernel, a Painlev ´ e equation, and a non trivial connection problem. W e expect that this p ersp ectiv e will be useful in other p ersistence and ﬁrst-passage problems where Pfaﬃan structures and in tegrable k ernels naturally arise. Finally , w e note that F redholm Pfaﬃans and P ainlev´ e equations also pla y a central role in the description of universal ﬂuctuations in growth pro cesses b elonging to the Kardar–P arisi–Zhang (KPZ) univ ersality class. In particular, one of the T racy–Widom distributions (the F 1 one), whic h gov erns height ﬂuctuations for curved initial condi- tions in several KPZ settings, admits a F redholm Pfaﬃan representation in volving the Airy k ernel and is asso ciated with a Painlev ´ e I I equation. A remark able feature of KPZ dynamics is the p ersistence of memory of the initial condition through a single curv ature parameter, which controls b oth the limiting dis- tribution and its scaling properties. In the present p ersistence problem, the exponent κ ( m ) plays an analogous role, enco ding the dep endence on the initial magnetization through the single parameter ξ = 1 − m 2 . F rom a structural viewp oin t, it is therefore natural to ask whether these parallels reﬂect a deep er connection. Since the Painlev ´ e VI equation go verning p ersistence is the most general member of the Painlev ´ e equations, and since low er Painlev ´ e equations suc h as Pain lev´ e I I arise in KPZ through well-kno wn conﬂuence limits, one may sp ec- ulate that suitable scaling or degeneration limits could relate p ersistence distributions to KPZ ﬂuctuation la ws. Exploring suc h connections lies b ey ond the scop e of the presen t work, but we b eliev e that the framework developed here provides a natural starting point for addressing this question. Statemen ts and declarations The authors hav e no relev ant ﬁnancial or non-ﬁnancial interests to disclose. The authors hav e no comp eting interests to declare that are relev ant to the conten t of this article. All authors certify that they hav e no aﬃliations with or inv olv ement in an y organiza- tion or en tity with an y ﬁnancial in terest or non-ﬁnancial interest in the sub ject matter or materials discussed in this man uscript. The authors ha ve no ﬁnancial or proprietary in terests in an y material discussed in this article. Data av ailabilit y statements The authors declare that the data supporting the ﬁndings of this study are av ailable within the pap er. Ac kno wledgemen ts During the man y y ears that lasted this persistence endeav or, the ﬁrst author has b eneﬁted from useful remarks, comments, or encouragemen t by many colleagues, in particular A.N. Boro din, C. Colleras, B. Derrida, P . Di F rancesco, G. Korchemsky , O. Liso vyy , J.-M. Maillard, K.T.R. McLaughlin, V. Pasquier and G. Sc hehr. 22 Both authors thank the referees and the handling editor for their careful read- ing and constructive suggestions, which ha ve help ed improv e the presentation of the man uscript. A The DHP formula for the p ersistence probabilit y as a Pfaﬃan gap-spacing probabilit y with the sec h k ernel One of the crucial p oin ts of the presen t pap er is to recognize that the formula [ 42 , Eq. (29) p. 773] for the p ersistence probability of Potts spins in the scaling r´ egime can b e recast as the follo wing one for ± Ising spins [ 45 , Eq. (8)]), P + 0 (  ; m ) : = 1 q p 0 ([ t 1 , t 2 ]; m )    1 /q = 1+ m 2 = 1 + m 2 D + + D − 2 + 1 − m 2 D + − D − 2 , (A1) b ecause in the random initial conditions one can iden tify a giv en colored Potts spin as a + spin with probability 1 /q = (1 + m ) / 2. By reversing the signs of all spins in the random initial condition (which does not aﬀect the subsequent dynamics) one also realizes that it holds that P + 0 (  ; − m ) ≡ P − 0 (  ; − m ) , (A2) where the P ± 0 (  ; m ) hav e the same meaning as in ( 1.1 ). As for our p 0 ([ t 1 , t 2 ]; q ), it is iden tical to the expression R ( q ; t 1 , t 2 ) of [ 42 , Eq. (29)]. In the notation deﬁned in ( 1.8 ), D ± = D ± (  ; 1 − m 2 ) are the F redholm determinants of, resp ectively , the even and odd parts of the sech kernel, with the corresp ondence for the indep endent v ariable 1 ≪ t 1 ≪ t 2 with  = log ( t 2 /t 1 ) > 0 , (A3) The pro of of the identit y ( A1 ) was sketc hed in the unpublished work [ 45 ] by the ﬁrst author. It just relies on applying the by now standard T racy-Widom technique [ 115 ] developed to study the orthogonal and symplectic ensembles of random matrix theory , which recasts a matrix Pfaﬃan F redholm determinan t in terms of tw o scalar functions link ed b y the Gaudin relation ( 2.50 ). The deriv ation of ( A1 ) is in fact v alid for an y even diﬀerence kernel acting on a symmetric interv al. In other words, it is a signature of the intrinsic nature of the Pfaﬃan p ersistence probabilit y . As to the existence of scaling limit in the particular case of the Ising mo del with m -magnetized random initial conditions, it has been pro ven since, in the rigorous probabilistic w ork [ 50 ]. All in all, this justiﬁes all the statements made in our Theorem ( 1.1 ), but for the ODE ( 1.9 ) satisﬁed b y the resolv ent for the sec h k ernel, which is established in Section 2.2 , and the v alue ( 1.4 ) of the persistence exp onen t, whic h comes from the low est- order term of the Boro din-Okounk o v form ula of App endix D for the geometric means of the F redholm determinan t D = D + . D − 23 B The P ainlev´ e VI solutions for the mean curv ature and the metric of Bonnet surfaces in R 3 In this Appendix, w e recall ho w the constitutiv e Gauss-Codazzi equations, whic h de- termine lo cally an ordinary surface immersed in the usual three-dimensional space, can b e glob al ly solv ed in terms of P VI transcenden ts for a particular class of isometry- preserving surfaces in tro duced and studied by the F renc h geometer Bonnet in 1867, and bearing his name since then. It was Bobenko and Eitner who found in 1994 that the mean curv ature for every Bonnet surface is in fact a certain P VI tau-function. Here w e presen t the P VI solution for their metric. Remark ably , we ﬁnd that not only the latter satisﬁes a rather esoteric sibling of the P VI equation that Chazy men tioned — without displa ying its solution! — as early as 1911 [ 26 ], but w e also sho w that it is related to the P VI determining the mean curv ature function by a certain folding transformation, of algebraic and non-birational nature. Of course, the extraordinary coincidence is that the ODEs satisﬁed b y the determi- nan ts giving the p ersistence probabilit y are but a particular case of the ones o ccurring in the Bonnet reduction of the Gauss-Co dazzi PDEs. This App endix is organized as follo ws. W e start by reviewing in a p edagogical w ay s ome classical material concerning the diﬀerential description of the geometry of ordinary (bi-dimensional) surfaces in (Euclidean space) R 3 . Along the wa y we review the nomenclature and w e set the appropriate notations needed for our purposes. The P VI solution for the mean curv ature function of Bonnet surfaces, based on [ 10 , 13 , 33 ], is recalled as Prop osition B.1 . The main nov elt y of this App endix, the P VI solution for the metric and its relation to the former one, is giv en as Proposition B.2 . Consider in the usual three-dimensional Euclidean space a smo oth enough surface Σ, th us determined — after Gauss — by t w o fundamental real quadratic forms I , I I deﬁned at each of its p oints r ∈ R 3 . W e take for granted the existence of a chart of complex conformal co ordinates ( z , z ) ≡ ( x + i y , x − i y ) whic h parametrize (at least lo cally) the domain Σ of R 2 of the corresp onding surface immersion r = r ( z , z ) in R 3 , so that these tw o quadratic forms can be equiv alen tly expressed as I : = d r · d r = q det [ g ij ]  d x 2 + d y 2  = G − 2 d z d z , (B1) I I : = − d r · d n = 1 2 J d z 2 + H d z d z + 1 2 J d z 2 , (B2) where the · in the deﬁnitions ab ov e represents the usual (Euclidean) scalar pro duct in R 3 . In the ﬁrst fundamen tal form, and b efore recalling what the (real-v alued) function H = H ( z , z ) and the (complex-v alued) one J = J ( z , z ) functions stand for in the sec- ond, we hav e used b oth (d x, d y ) and (d z , d z ) viewp oints, assuming for the latter that the conformal co ordinates are also isothermal . This means that the p ositiv e deﬁnite 2 × 2 matrix [ g ij ] giving the metric is diagonal. Con venien tly for what follo ws, w e ex- press its sole non-zero and real-v alued co eﬃcien t as g ij ≡ δ ij / G 2 , with G 2 = G 2 ( z , z ). It also turns out that the existence of suc h isothermal co ordinates is alwa ys ensured for 24 Bonnet surfaces, except at their single [ 101 ] “umbilic” p oint (where the t wo principal curv atures coincide — more on this around ( B8 ) b elo w). The function G , whic h in the tw o fundamen tal quadratic forms only app ears through the square of its inv erse, is the c onformal factor for the metric. Indeed, when one makes a reparametrization z  → w , z = f ( w ) of the surface c hart, with f ana- lytic and f ′  = 0 within the domain f − 1 (Σ), the piece 1 / G 2 transforms as | f ′ ( w ) | 2 / G 2 . Bearing in mind these relationships, w e shall indiﬀeren tly refer to either 1 / G 2 , G 2 , or G , as the conformal factor for the metric, the metric factor, or even simply the metric. As for the second fundamental form ( B2 ) (not necessarily a p ositiv e deﬁnite one), it in volv es the oriented unit v ector n normal to the tangen t plane spanned by ( ∂ r , ∂ r ), so that equiv alently I I = d 2 r · n . Here and elsewhere, partial deriv ativ es with resp ect to the conformal co ordinates are deﬁned and abbreviated as ∂ ≡ ∂ z : = 1 2 ( ∂ x − i ∂ y ) , ∂ ≡ ∂ z : = 1 2 ( ∂ x + i ∂ y ) . The necessary conditions that the three co eﬃcien ts ( G , H , J ) whic h app ear in the tw o fundamen tal quadratic forms hav e to satisfy (as functions of z , z ) to lo c al ly describ e a surface date back to the nineteenth cen tury , when they were ﬁrst written do wn by Gauss and Codazzi (along with and independently b y P eterson and Mainardi for the second ones [ 94 ]), to wit 4 ∂ ∂ (log G ) = G − 2 H 2 − G 2 |J | 2 (B3) G ∂ J = G − 1 ∂ H and G ∂ J = G − 1 ∂ H , its complex conjugate . (B4) The other real-v alued function H = H ( z , z ) that appears in the abov e under- determined system of non-linearly coupled PDEs is the me an curvatur e H : = κ 1 + κ 2 2 , that is to say the arithmetic mean of its t wo princip al curvatur es κ 1 , κ 2 . The latter are deﬁned at each p oin t of the surface as the tw o eigenv alues — assumed non- degenerate — of the so-called W eingarten’s “shap e op erator”. This linear op erator (living abstractly in the tangent space to the surface) can b e represented by a 2 × 2 real symmetric matrix, simply built from those asso ciated to the tw o fundamental quadratic forms in the (d x, d y ) basis [ I I ] × [ I ] − 1 =  H + G 2 Re J −G 2 Im J −G 2 Im J H − G 2 Re J  . (B5) By con truction, the trace of this shape op erator is 2 H = κ 1 + κ 2 , t wice the mean curv ature, while its determinan t deﬁnes Gauss — or total — curv ature K : = κ 1 κ 2 = det [ I I ] / det [ I ] = H 2 − G 4 |J | 2 . (B6) That quan tity is not only in v ariant under conformal reparametrization, but it is also intrinsic . Indeed, from the ﬁrst of the Gauss-Codazzi equations, ( B3 ), one also reads that K = 4 G 2 ∂ ∂ (log G ) , (B7) where the right-hand-side in volv es the Laplace-Beltrami op erator for the induced Rie- mannian manifold (here tw o-dimensional). That the total curv ature dep ends in fact 25 only on the metric of the surface without the need for any R 3 -em b edding is the gist of Gauss celebrated The or ema e gr e gium (“remark able theorem”). The third co eﬃcient J = J ( z , z ) which appears in the second fundamental form and in the Gauss-Co dazzi equations is deﬁned through a conformally-inv arian t (2 , 0) t wo-form J d z 2 : =  ∂ 2 r · n  d z 2 called the quadratic Hopf diﬀerential. Henceforth w e shall simply refer to J = J ( z , z ) as the Hopf factor . Contrarily to G and H , it is generically a complex-v alued ob ject, its mo dulus |J | b eing alwa ys nonzero a w ay from the so-called umbilic p oints of the surface, where (b y deﬁnition) the tw o principal curv atures coincide. Indeed, by using the expression ( B6 ) for the determinant of the shap e operator and combining it with the deﬁnitions for the mean and Gaussian curv atures, a string of (trivial) identities pro duces ev entually the (square of ) the so- called skew curvatur e L , that is to sa y (half ) the diﬀerence betw een the t w o principal ones G 4 |J | 2 = H 2 − K =  κ 1 + κ 2 2  2 − κ 1 κ 2 =  κ 1 − κ 2 2  2 ≡ L 2 , L : = κ 1 − κ 2 2 . (B8) Before addressing the historical Bonnet problem for which the complex-v alued nature of the Hopf factor pla ys a key rˆ ole, let us in tro duce for a generic surface Σ another natural quantit y that one can deﬁne using ( B8 ), viz. its Wil lmor e ener gy E = E [Σ]. Its consideration will b e v ery helpful later on to relate our F redholm determinants solutions of the Gauss-Co dazzi equations to glob al properties of Bonnet surfaces. T o wit, observe that by integrating ( B8 ) ov er the whole surface Σ (or part of it) with the tw o-form area d A : = q det [ g ij ] d x ∧ d y ≡ G − 2 dRe z ∧ dIm z , (B9) one obtains by construction a non-negative and conformally inv ariant global measure of the surface asphericity E : = Z Σ d A  H 2 − K  = Z Σ ( dRe z ∧ dIm z ) G 2 |J | 2 , (B10) since by ( B8 ) the in tegrand is lo cally non-zero if a nd only if κ 1  = κ 2 . In fact, this Willmore energy is a fundamen tal quan tity , whic h app ears time and again and in one guise or in another in v arious domains of the natural sciences. W e brieﬂy mention a couple of examples b elo w. Let us ﬁrst remark that ( B10 ) inv olves Euler’s char acteristic , deﬁned in the con- tin uous context here b y Gauss in tegral curv ature, viz. X E : = (2 π ) − 1 R Σ d A K . F or compact surfaces of ﬁxed genus, this constant top ological term is usually discarded to deal with the shifted (and still non-negative) Willmore functional e E : = Z Σ d A H 2 ≡ E + 2 π X E , (B11) 26 its one-dimensional reduction e E  → R d l κ 2 ( l = arc length) b eing nothing but Euler- Bernouilli’s elastic a (see, e.g., [ 84 , 89 ], and references therein)! As for this famous problem, a long-standing issue for geometers and whic h is still of interest now adays [ 110 , 111 ] has b een to globally minimize that “conformal area” e E by v arying H . It turns out that this very same quan tity also sho ws up in models of biological membranes, where e E represents the so-called (Canham-)Helfric h free energy [ 24 , 63 ] and even in string theory , where Poly ak ov [ 96 ] considered it as the extrinsic action for tw o-dimensional quan tum gra vity . This b eing said, a classical theorem by Bonnet ensures that any solution ( G , H , J ) to the Gauss-Codazzi equations ( B3 )–( B4 ) determines a unique surface in R 3 (this of course lo cally , and up to “rigid motion”, i.e. up to arbitrary ﬁnite translations and rotations). F or instance, the so-called constant mean curv ature surfaces are obtained with H = const, and their metric ob eys the Liouville PDE (if this const = 0 as for minimal surfaces), or the (b y no w!) classically in tegrable sin(h)-Gordon PDE (if that const  = 0). Con versely , one can w onder whether there is any redundancy in the Gauss-Co dazzi equations, that is to say whic h data among the three functions ( G , H , J ) can b e disp ensed of in the nonlinear system of under-determined PDEs they obey , and what family of surfaces in R 3 w ould thereby b e deﬁned. F ollo wing this geometric line of though t, Bonnet raised and answ ered in his 1867’s seminal work the problem that still b ears his name: Given a real surface in R 3 , how to determine all surfaces whic h are applic able on it, that is to say whic h p ossess (up to conformal transformations) the same t wo principal curv atures κ 1 , κ 2 ? Since isometries of R 3 conserv e the metric hence in particular the Gaussian curv ature κ 1 κ 2 , these considerations led Bonnet to the disco very and the characterization of a family of isometry-pr eserving surfaces, all ha ving the same — non-constant — mean curv ature function. T ec hnically , these Bonnet surfaces are obtained by eliminating from the Gauss- Co dazzi equations the phase of the complex-v alued Hopf factor J . After an appropri- ate conformal transformation (more on this b elo w), and up to an arbitrary choice of the origin of the lo cal co ordinates, one ﬁnds that the sole nov el surfaces, real-v alued and analytic, are of three diﬀeren t t yp es (App endix B of [ 33 ] pro viding a concise but exhaustiv e summary). This classiﬁcation essentially depends on whether a certain real parameter — purp osefully denoted here ν 2 , as in the kernel ( 2.38 ) — is negative, p os- itiv e, or zero, three cases referred to muc h later [ 25 ] by ´ E. Cartan as resp ectiv ely A, B, and C. With that ν b eing therefore either purely imaginary or real, let us deﬁne the follo wing complex-v alued function of ( z , z ), the second equiv alent formula b eing mere (!) trigonometry: J ν 2 ( z , z ) : = ν coth ( ν z / 2 ) − ν coth ( ν Re z ) ≡ ν sinh ( ν Re z ) sinh ( ν z / 2 ) sinh ( ν z / 2 ) . (B12) One of the b eneﬁts of this algebraic expression is to treat all cases on the same fo oting: the hyperb olic lines simply become ordinary trigonometric ones when goes from ν 2 > 0 27 to ν 2 < 0, with the understanding that the remaining v alue ν = 0 (case C in Cartan’s classiﬁcation) at the intersection of these tw o conditions can b e obtained by passing to the (well-deﬁned) “rational” limit ν → 0 in the formula abov e (so that J 0 ( z , z ) = 2 z − 2 z + z ), and all subsequent ones. Of course, to be able to use that function J ν 2 ( z , z ) as the Hopf factor with a prop erly deﬁned domain for the conformal v ariables, the denominators in ( B12 ) hav e to be alwa ys nonzero. One thereb y gets t w o admissible regions, separated by and symmetric with resp ect to the imaginary axis Re z = 0. These tw o mirr or regions consist into a pair of inﬁnite strips (degenerating into the right and left half-plane when ν 2 → 0), whic h are vertical or horizontal ones dep ending whether ν 2 is negative or p ositiv e, and for which the mo dulus/the absolute v alue | ν | = √ ∓ ν 2 ( ∓ in case A/B) serv es as a scale factor for their width or heigh t. As for the conformal conten t of ( B12 ) brieﬂy alluded to ab o ve, since the imaginary part of the Hopf factor is that of an analytic function, it alwa ys satisﬁes lo cally within eac h of those strips 4 ∂ ∂ Im J ν 2 = 0, the bi-dimensional Laplace equation. This lo cal, harmonic characterization is in fact a glob al one, if one recalls (or b ecomes aw are of ) the classical result of [ 118 ], although obtained in a completely diﬀerent analytic con text (we refer to [ 33 ], p. 23, for another pro of using PDEs metho ds in the original, geometric language). Summarizing, either of these arguments shows that the expression ( B12 ) for the Hopf factor can b e considered as essen tially deﬁned in a glob al and unique wa y (see [ 118 ] for details), notably for probabilistic reasons dictated b y the underlying rep- resen tation through the Poisson kernel of the non-negativ e solutions to the Laplace equation in a strip (or in any conformally equiv alen t domain). A consequence of this harmonic b eha vior is that on either of those strips (or on b oth half-spaces) it holds that 2 ∂ J ν 2 ( z , z ) = 2 ∂ J ν 2 ( z , z ) = |J ν 2 ( z , z ) | 2 , (B13) where the squared mo dulus of the Hopf factor ab ov e simply ev aluates to |J ν 2 ( z , z ) | 2 =  ν sinh ( ν Re z )  2 , (B14) since J ν 2 ( z , z ) = J ν 2 ( z , z ) b y the very deﬁnition ( B12 ) and the fact that ν 2 is alwa ys real. F or reference, we also men tion that the argumen t of the Hopf factor is giv en by the nice expression tan  arg J ν 2 2  = − coth  ν Re z 2  tan  ν Im z 2  (B15) already presen t in Bonnet’s original w ork. The output of all this is that if one tak es for the Hopf factor J precisely the function J ν 2 giv en b y ( B12 ), or a m ultiple of it and/or of its argumen ts, the tw o Codazzi equations ( B4 ) automatically reduce to a single one, and this in turn implies that both 28 the mean curv ature function and the conformal metric factor can b e c hosen to dep end solely on Re z according to the piv otal relationship (see, e.g., equation (21.d) in [ 33 ]), d H (Re z ) dRe z =  ν sinh ( ν Re z )  2 G 2 (Re z ) . (B16) Inciden tally — and this will b e a crucial observ ation for the p ersistence problem — the right-hand-side ab ov e is nothing but the Willmore “elastic energy” densit y ( B10 ). The Gauss-Co dazzi PDEs ( B3 )–( B4 ) for this class of isometry-pr eserving surfac es — realized b y the contin uous deformation from a representativ e one as Im z is v ar- ied — thereby reduce to a pair of coupled nonlinear ﬁrst and second-order or dinary diﬀeren tial equations (ODEs), H ′ = |J ν 2 | 2 G 2 , (B17) 1 2  log G 2  ′′ = G − 2 H 2 − |J ν 2 | 2 G 2 , (B18) the primes ab o ve standing for diﬀerentiation with resp ect to the (sole) indep enden t v ariable Re z of this system. Note also that we hav e contin ued to employ the same sym b ols G = G (Re z ) and H = H (Re z ) for the metric and mean curv ature functions of these Bonnet surfaces. The use of Re z (instead of x ) for their independent v ari- able will also b e fav ored, notably b ecause that notation serves as a reminder of the bidimensional nature of the conformal v ariables ( z , z ) in the initial Gauss-Co dazzi PDEs. Eliminating G 2 b et w een ( B17 ) and ( B18 ), one immediately reco v ers the third-order nonlinear ODE 1 2  H ′′ H ′  ′ + H ′ −  ν sinh ( ν Re z )  2 H 2 + H ′ H ′ = 0 (B19) that Bonnet found in 1867 ([ 15 ], equation (52) p. 84) for the mean curv ature function of “his” surfaces. It is displa yed here follo wing the standardization of [ 34 ] (equation (B.1) p. 289). The w orks [ 11 ] (equation (25) p. 55) and [ 13 ] (equation (3.31) p. 32) ha ve a diﬀeren t normalization for the conformal coordinates and the Hopf factor in the corresp onding ( B13 )–( B16 ), implying in particular that the sign of their H ′ is alw a ys negativ e, thus opp osite to ours, cf. ( B16 ) here. A t the end of the 19th century , Hazzidakis [ 62 ] found for that third-order ODE ( B19 ) a “ﬁrst integral”,  H ′′ 2 H ′ + ν coth ( ν Re z )  2 + H ′ +  ν sinh ( ν Re z )  2 H 2 H ′ + 2 ν coth ( ν Re z ) H = ν 2 θ 2 , (B20) where θ 2 is the corresp onding integration constant, an alw ays real y et p ossibly negative quan tity , i.e. θ can be purely imaginary , at par with our con ven tion for ν 2 . Note that a simple means of c hecking ( B20 ) is to multiply ( B19 ) b y the integrating factor 29 H ′′ / H ′ + 2 ν coth ( ν Re z ), or conv ersely to diﬀerentiate (once exhibited!) the second- order equation ( B20 ): ( B19 ) w ould app ear, up to that o v erall m ultiplying factor. Once the (arbitrary) origin of the conformal coordinates is ﬁxed, one can c heck using ( B16 )–( B20 ) that the mean curv ature Re z 7→ H (Re z ) and the metric factor Re z 7→ G (Re z ) are resp ectively o dd and ev en. Ov erall signs or scaling factors for these tw o real functions deﬁned on R \ { 0 } are otherwise arbitrary and can b e c hanged at will (for generic conv entions, see, e.g., [ 32 ]) as long the asso ciated joint equality ( B16 ) is maintained. One thereby essentially obtains the same r e al Bonnet surface, whic h in general depends on six parameters [ 15 , 27 , 33 ], and w e shall alw ays consider represen tatives diﬀering by such normalizations as equiv alent. Nev ertheless, b eyond a local description, the existence of the global solution to the strongly nonlinear ODE ( B20 ) and its prop erties has b een considered for many years b y the best geometers [ 15 , 25 , 27 ] as a diﬃcult if not insurmoun table problem. Ev entually in 1994 [ 11 ], Bob enk o and Eitner succeeded in identifying ( B19 )–( B20 ) as a certain codimension-one P VI tau-function, the sole real parameter θ 2 for the latter b eing determined by Hazzidakis’ ﬁrst in tegral. Their key observ ation was to recognize in the W eingarten moving frame equations an appropriately-gauged Jim bo-Miwa 2 × 2 matrix Lax pair. This follo wed the pioneering work [ 10 ] b y the ﬁrst of these authors, where concepts and methods from integrable systems and soliton theory were fruitfully imp orted into diﬀeren tial geometry (see also [ 72 ] for related ideas). A notew orthy consequence of this crucial corresp ondence — whic h took therefore more than a century to be un v eiled — is that the global existence of a Bonnet surface is no w ensured — at least conceptually! In practice see App endix D — through Jimbo’s famous solution of the tau-function P VI connection problem [ 68 ]. Since this remark able disco very , the mapping of the mean curv ature function of Bonnet surfaces to a particular instance (see ( B24 )) of what is called in the mo dern literature the Ok amoto–Jimbo-Miwa sigma-form of P VI (although it ﬁrst appeared as expression t p. 341 in [ 26 ] has b een detailed at length in articles, lecture notes, or even b ooks [ 11 , 13 , 14 , 34 ]. W e also mention that the extrap olation to the generic four- parameter P VI giv es, due to its in trinsic geometric origin, probably the “b est” Lax pair for P VI [ 32 , 33 ]. Some prop erties of the Bonnet P VI mean curv ature function are recalled in the (long) Prop osition B.1 b elow. This is mainly to set the stage for our Proposition B.2 , the core of this Appendix, whic h addresses the P VI solution of the second-order ODE satisﬁed b y the Bonnet metric function G = G (Re z ) itself  G ′′ − 2 ν 2 G 3 − ν 2 θ 2 G  2 + ( 2 ν coth ( ν Re z ) G ) 2  G ′ 2 − ν 2 G 4 − ν 2 θ 2 G 2  = 0 . (B21) Although it is immediate to obtain this equation by eliminating the mean curv ature function H (and its deriv ativ es H ′ , H ′′ ) b etw een ( B17 ), ( B18 ), and ( B20 ), to the b est of our kno wledge ( B21 ) do es not seem to ha ve been written down before. One of the reasons might b e that ( B21 ) has to b e identiﬁed as a particular incarnation in this geometric context of the rather esoteric Chazy C VI equation, of which the general solution w as made explicit only in 2006 [ 36 ]. 30 It turns out that an equation such as ( B21 ) is in fact [ 34 ] “half-w ay” b et ween the usual P VI equation and its sigma-form/tau-function, since it can also b e view ed as the second-order second-degree nonlinear ODE satisﬁed by a properly shifted and rescaled impulsion in Ok amoto’s P VI Hamiltonian formalism (the solution of C VI is recalled in App endix C ). Last but not least, there is also an additional lay er of non-trivial Painlev ´ e rela- tionships holding here due to the sp eciﬁc mono drom y exp onen ts of the Bonnet family . Indeed, it happ ens that the solution to ( B21 ) can also b e expressed algebraically in terms of the Bonnet-P VI mean curv ature function through a so-called folding P VI transformation, here a quadr atic one. This ev en more recondite P VI symmetry is presen t only under a certain co dimension-t w o constraint on the four mono drom y exp onen ts, namely when either t wo of them are zero or when there exists tw o couples of pairwise equal ones. This symmetry is in fact a generalization at the lev el of P VI of the so-called Goursat trans- formation for Gauss h yp ergeometric function 2 F 1 ( a, b, c ; x ) when the parameters and the independent v ariables are constrained according to 2 F 1  a, b, a + b + 1 2 ; x  = 2 F 1  a 2 , b 2 , a + b + 1 2 ; 4 x (1 − x )  . (B22) The folding quadratic transformation for P VI w as ﬁrst found (indep enden tly) by Kitaev and Manin in the 1990’s, until it was generalized and extended for all Painlev ´ e transcenden ts and recast using metho ds of algebraic geometry by Tsuda, Ok amoto, and Sak ai in [ 117 ]. The most general formulas for these quadratic and the quartic P VI transformations are recalled in Appendix C , along with their Hamiltonian signiﬁcation as generalized time-dep endent canonical transformations. Finally , recall that Bonnet surfaces are so-called W eingarten surfaces [ 98 ] since the mean curv ature and the metric are related by ( B16 ), one has d G ∧ d H = 0. One of the b y-pro ducts of our ﬁndings is therefore to provide explicit relationships showing ho w this is realized at the P VI lev el. Prop osition B.1. The me an curvatur e of Bonnet surfac es as a P VI tau- function (Bob enko-Eitner [ 11 , 13 ], se e also [ 33 ]) F or al l typ es of P VI Bonnet surfac es as p ar ametrize d by ν 2 and θ 2 , the glob al solution to t heir me an curvatur e function ob eying ( B20 ) c an b e obtaine d thr ough H (Re z ) = 2 εν h θ 2 ( s ε ( ν Re z )) , s ε ( ν Re z ) : = 1 − ε coth ( ν Re z ) 2 , ε 2 = 1 . (B23) With r esp e ct to either br anch ε = ± 1 of the indep endent variable s = s ε ( ν Re z ) , the function s 7→ h = h θ 2 ( s ) ob eys the se c ond-or der se c ond-de gr e e nonline ar ODE  s ( s − 1) d 2 h d s 2  2 + 4 d h d s  s d h d s − h   ( s − 1) d h d s − h  − θ 2  d h d s  2 = 0 . (B24) 31 On either symmetric admissible r e gion of the c onformal c o or dinates, this deﬁnes two mirr or-like Bonnet surfac es H ( − Re z ) = −H (Re z ) having the same metric G 2 (Re z ) = d h θ 2 ( s ) d s    s = s ε ( ν Re z ) . (B25) This r e duc e d me an curvatur e function c oincides with an auxiliary Okamoto Hamil- tonian evaluate d on the solutions q = q ( s ) , p = p ( s ) of the e quations of motion for a P VI Hamiltonian H VI ( p, q , s ) with mono dr omy exp onents { θ α , θ β , θ γ , θ δ } : h ( s ) = s ( s − 1) H VI ( p ( s ) , q ( s ) , s ) , d q d s = ∂ H VI ∂ p , d p d s = − ∂ H VI ∂ q . (B26) This identiﬁc ation h ≡ h θ 2 ≡ h takes plac e for a c o-dimension one family of mon- o dr omy exp onents having any signe d values among the two sets i) or ii) p ar ameterize d by θ 2 :  θ 2 α , θ 2 β , θ 2 γ , ( θ δ − 1) 2  = ( ( θ 2 , 0 , 0 , θ 2 ) set i) (0 , θ 2 , θ 2 , 0) set ii) (B27) The bir ational e quivalenc e existing b etwe en ( s, h, d h d s ) and the P VI function q = q ( s ) is simpler for the set ii) h q − s = − d h d s , h = h θ 2 ( s ) , (B28) wher e that q = q θ 2 ( s ) solves the P VI e quation with c o eﬃcients that ar e even in θ , as the e quation for the r e duc e d me an curvatur e: [ α, β , γ , δ ] = " θ 2 α 2 , − θ 2 β 2 , θ 2 γ 2 , 1 − θ 2 δ 2 # ( B 27 ) . ii) =  0 , − θ 2 2 , θ 2 2 , 0  . (B29) Conversely, given a br anch of that P VI function s 7→ q θ 2 ( s ) , ther e exists an ex- pr ession of h = h θ 2 ( s ) in terms of the lo garithmic derivative of the Chazy-Malmquist tau-function τ = τ θ 2 ( s ) , whose unique movable singularity is a simple p ole of r esidue unity, and which is even in b oth the ﬁrst-or der derivative of q θ 2 ( s ) and in the mono dr omy exp onent θ : h θ 2 ( s ) = s ( s − 1) d log τ θ 2 ( s ) d s = 1 4 s 2 ( s − 1) 2 q θ 2 ( q θ 2 − 1)( q θ 2 − s )  d q θ 2 d s  2 − θ 2 4 q θ 2 − s q θ 2 ( q θ 2 − 1) . (B30) Since the conten ts of Prop osition B.1 are well kno wn, we recall some elements of its pro of after asserting Proposition B.2 . This is mainly to pa ve the ground for the demonstration of the latter, where the P VI solution for the metric factor of Bonnet surfaces is exhibited. 32 Prop osition B.2. A Chazy C VI solution for the metric factor of P VI Bonnet surfac es The solution of the ODE ( B21 ) giving the metric of Bonnet surfac es c an b e dir e ctly expr esse d as G 2 (Re z ) = − g 2 θ 2 ( t ( ν Re z )) , (B31) wher e the indep endent variable t = t ( ν Re z ) for this one-p ar ameter Chazy C VI func- tion g = g θ 2 ( t ) in r ational form is also the one for a P VI function Q ( t ) with the “folde d” set of mono dr omy exp onents { θ α , θ β , θ γ , θ δ } =  1 − θ 2 , 1 − θ 2 , θ 2 , θ 2  , (B32) and a c onjugate impulsion P ( t ) , so that g θ 2 ( t ) = 2 P ( t ) ( Q ( t ) − 1 ) ( Q ( t ) − t ) t − 1 ≡ t Q ( t ) d Q ( t ) d t − 1 2 − 1 − θ 2( t − 1)  Q ( t ) − t Q ( t )  . (B33) This P VI solution Q ( t ) is determine d by a quadr atic tr ansformation which inter- twines it algebr aic al ly but never bir ational ly with any of the admissible q ( s ) ( B27 ) for the me an curvatur e, with in p articular for their r esp e ctive indep endent variables s = s ε ( ν Re z ) (on either br anch ε 2 = 1 ) and t = t ± 1 ( ν Re z ) (for either sign ± ) s = 1 2 + 1 4  √ t + 1 √ t  , henc e t ( ν Re z ) = coth 2  ν Re z 2  or tanh 2  ν Re z 2  . (B34) The functions giving the r e duc e d me an curvatur e and the metric factor for the Bonnet surfac es thus ob ey d h θ 2 ( s ) d s    s = s ε ( ν Re z ) = − g 2 θ 2 ( t )    t = t ± 1 ( ν Re z ) . (B35) Pr o of (of the Proposition B.1 ). W e start by justifying the app earance of the sign ε = ± 1 in ( B23 ), which comes after c hanging ν  → εν in the reduced Gauss-Co dazzi equations, or in the resulting Bonnet- Hazzidakis ones ( B17 )–( B20 ). Since all these equations are even in ν , they are of course in v ariant under this ad ho c op eration. Y et, the explicit in troduction of that sign is a simple means of keeping trac k of the existence of the tw o fundamen tal regions for Re z originating from the meromorphic “barrier” in ( B12 ), and which resurfaces here through the tw o branches in the mean curv ature P VI indep enden t v ariable s = s ε . Indeed, b ecause of the o ddness of the coth ( · ) function, this underlying “mirror” symmetry can be realized as the inv olution s ε ( ν Re z ) = 1 − coth ( εν Re z ) 2 = 1 − ε coth ( ν Re z ) 2 ≡ 1 − s ε ( − ν Re z ) ≡ 1 − s − ε ( ν Re z ) . (B36) Equiv alently , one could switch (or not) these t wo branches changing ν  → − ν and/or Re z  → − Re z , while keeping the product ν Re z even or o dd. Keeping that ε as a remaining degree of freedom will be v ery useful to ac hiev e in the con text of the persistence problem a prop er corresp ondence betw een geometric quan tities and probabilistic ones. 33 Note that another w a y of pro ceeding w ould consist of deﬁning from the onset the Hopf factor ( B12 ) J ν 2  → J ε,ν 2 b y “pulling out” an explicit sign ε 2 = 1, for instance according to J ε,ν 2 ( z , z ) = ε ( ν coth ( ν z / 2) − ν coth ( ν Re z )) . (B37) This would mo dify the fundamental relationship ( B17 ) as ε H ′ = |J ν 2 | 2 G 2 , so that an ad- ditional sign would app ear in the Bonnet and Hazzidakis equations, multiplying all the o dd-terms in the mean curv ature function and its deriv ativ es. (Incidentally , Bob enko and co work ers made suc h a c hoice with ε = − 1, so that they alwa ys hav e their H ′ < 0.) T o ha ve the simplest p ossible displa yed equations, we stick to our original prescription, bearing simply in mind that whatever ν , the mean curv ature function H (Re z ) has to b e o dd with resp ect to Re z . A t an y rate, with the explicit change of the indep enden t v ariables ( B23 ), one has d s ε ( ν Re z ) dRe z = εν 2 sinh 2 ( ν Re z ) ≡ 1 2 εν |J ν 2 (Re z ) | 2 , (B38) so that b oth parameters ε and ν , along with the mo dulus of the Hopf factor are absorb ed in the single Co dazzi equation ( B16 ) when one changes v ariables and functions (Re z , H )  → ( s, h ). Hence one immediately obtains ( B25 ), for short G 2 = h ′ , primes standing here and henceforth for s -deriv atives. Plugging that result for the metric in Gauss equation ( B18 ) gives for h = h ( s ) the third-order ODE s ( s − 1)  h ′  s ( s − 1) h ′′  ′ − s ( s − 1) h ′′ 2  + 2 h ′  s ( s − 1) h ′ 2 − h 2  = 0 . (B39) Con verting into these v ariables Hazzidakis’ integrating factor ( ≡ − 8 ν 3 s 2 ( s − 1) 2 h ′′ ) for Bonnet equation ( B19 ), one c hecks that the left-hand-side of ( B 39 ) = h ′ 3 2 h ′′ "  s ( s − 1) h ′′ h ′  2 +  2 h − (2 s − 1) h ′  2 h ′ − h ′ # ′ . (B40) On the one hand, if we denote by θ 2 the ﬁrst integral (p ossibly negative, as for ν 2 ) of the total s -deriv ativ e ab o ve on the non-singular solutions ([ 92 ], Remark 1.1, p. 350) where that parameter-dep enden t function h = h θ 2 ( s ) is neither constant nor aﬃne (i.e. when the pro duct h ′ h ′′  = 0), we therefore obtain  s ( s − 1) h ′′ h ′  2 +  2 h − (2 s − 1) h ′  2 h ′ − h ′ = θ 2 . (B41) This expression is equiv alent to ( B24 ) after some straigh tforward algebraic rearrangement. On the other hand, from Ok amoto’s theory (see App endix C for a digest), one kno ws that for arbitrary mono drom y exp onen ts the auxiliary P VI Hamiltonian ev aluated on the equations of motion, namely the function s 7→ h ( s ) = s ( s − 1) H VI ( p ( s ) , q ( s ) , s ) + some aﬃne shift in s, (B42) ob eys a certain second-order second-degree nonlinear ODE (denoted E VI in [ 91 ], recalled as ( C3 ) in Appendix C ), that one can also view as a polynomial in h ′ . Iden tifying its coeﬃcients, one veriﬁes that the θ 2 -parameter-dep enden t reduced mean curv ature function h ≡ h θ 2 solving ( B24 ) do es coincide with the auxiliary Hamiltonian function h deﬁned through ( B42 ) when the mono drom y exp onen ts  θ α , θ β , θ γ , θ δ  b elong to either set i) or ii) display ed in ( B27 ) whatever the signs chosen for the squares o ccurring there, while the aﬃne function of s in ( B42 ) is iden tically zero in both cases (see Prop osition C for details). 34 F rom now on, we shall most of the time simply write h for this θ 2 -parameter-dep enden t reduced mean curv ature Bonnet function h θ 2 , which is also equal to that auxiliary P VI Hamil- tonian h with either set i) or ii) or mono dromy exponents. F or both sets, note that their P VI co eﬃcien ts satisfy β + γ = 0: this in volution { s, q ( s ) , h ( s ) } ↔ { 1 − s, 1 − q (1 − s ) , − h (1 − s ) } is realized changing the sign ε in s = s ε ( ν Re z ), cf. ( B36 ): it corresp onds to a reversal of orien tation for these tw o mirror Bonnet surfaces, lea ving their metric in v ariant. As for the justiﬁcation of ( B28 ), one knows that for arbitrary  θ α , θ β , θ γ , θ δ  the bira- tional equiv alence ( s, h, h ′ )  → q has a rather bulky expression (see T able R in [ 93 ], repeated as (34) in [ 33 ], or as (B.54) in [ 34 ]). Y et, for the second set ( B27 ) whic h has θ δ = 1, a lot of simpliﬁcations take place, and one obtains — indep endently of θ 2 , and still assuming h ′  = 0 — the very simple ( B28 ), where q solv es P VI with co eﬃcien ts ( B29 ). Concerning the penultimate p oin t, the tau-function formula ( B30 ) for h comes from the deﬁnition of the auxiliary Hamiltonian, using the expression of the impulsion p = p ( s ) in terms of the “v elo cit y” d q d s determined by one of Hamilton’s equation of motion d q d s = ∂ H VI ∂ p . Indeed, with the speciﬁc quadratic dep endence in p of the polynomial Hamiltonian ( C8 ), the follo wing simple aﬃne relationship holds, d q d s = q ( q − 1)( q − s ) s ( s − 1)  2 p − θ β q − θ γ q − 1 − θ δ − 1 q − s  , (B43) where, by the deﬁnition of the P VI co eﬃcien ts in terms of the mono drom y exponents (cf. ( B29 )), neither q = q ( s ; [ α, β , γ , δ ]) nor its deriv ative dep end on the signs of the  θ α , θ β , θ γ , θ δ  , while the conjugate impulsion p = p ( s ) giv en b y the solution of ( B43 ) ob viously depends explicitly on those chosen for the last three ones. W e momen tarily highligh t this dependence by writing p { θ β ,θ γ ,θ δ } for the impulsion p = p ( s ) solution of ( B43 ), hence such that p { θ β ,θ γ ,θ δ } : = 1 2 s ( s − 1) q ( q − 1)( q − s ) d q d s + 1 2  θ β q + θ γ q − 1 + θ δ − 1 q − s  . (B44) Lo oking up [ 34 ], equations (B.42) and (B.44), one reads that the logarithmic deriv ative of the so-called there Chazy-Malmquist tau-function τ ≡ τ VI , C ,x can b e expressed in general as s ( s − 1) d log τ d s = p { θ β ,θ γ ,θ δ } p {− θ β , − θ γ , 2 − θ δ } q ( q − 1)( q − s ) + . . . , (B45) where the omitted terms . . . are identically zero for the sets i) and ii). As for the corresp on- dence of notations, our signed  θ α , θ β , θ γ , θ δ  are (in that order) the { θ ∞ , θ 0 , θ 1 , θ x } of [ 34 ], with the indep enden t P VI v ariable denoted there x (here we ha ve been using s ), while the Riccati factor is R ( θ 0 , θ 1 , θ x ) ≡ 2 p { θ β ,θ γ ,θ δ } q ( q − 1)( q − s ) in terms of our impulsion ( B44 ). W e sp ecialize the ab ov e result for the Bonnet family , switc hing also for a while to notations emphasizing the dep endence on and the parit y with resp ect to the mono drom y exponent θ . Let us therefore rename the P VI function for set ii) as q θ 2 = q ( s ; h 0 , − θ 2 / 2 , θ 2 / 2 , 0 i ). F or that same P VI function q θ 2 , if one c hooses appropriate relative signs when expressing the t w o mono drom y exp onen ts θ β , θ γ in terms of θ (still setting in ( B44 ) θ δ = 1), there exists tw o conjugate impulsions p ± θ , which can be obtained by simply rev ersing the sign of θ . Namely , if one deﬁnes p ± θ : = p { θ β ,θ γ ,θ δ }    θ β = ∓ θ,θ γ = ± θ,θ δ =1 = 1 2 s ( s − 1) q θ 2 ( q θ 2 − 1)( q θ 2 − s )  d q θ 2 d s ± θ q θ 2 − s s ( s − 1)  , (B46) then equation ( B45 ) for the Bonnet-P VI tau function τ ≡ τ θ 2 reduces to s ( s − 1) d log τ θ 2 d s = p + θ p − θ q θ 2 ( q θ 2 − 1)( q θ 2 − s ) , (B47) 35 so that ( B30 ) follo ws immediately , our deriv ation also explaining why that form ula appears under a factorized form even tually ev en in both d q d s and θ . Note also that the case θ = 0 is simply obtained by taking the limit θ → 0 (all formulas b eing holomorphic in θ ), the tw o conjugate impulsions p ± θ thereb y coalescing in to a single one p 0 . Finally , the fact that this tau function has only one mo v able simple p ole s i with residue unit y , i.e. d log τ θ 2 d s ∼ 1 s − s i , is ensured by the general construction put forward b y Painlev ´ e and his school (see for instance section B.V of [ 34 ] for a recap), the lo cation s i of this p ole in the complex plane depending in general on the initial conditions. In our probabilistic context where the mean curv ature is essentially the resolv ent of an integral k ernel K θ,ν generating a determinan tal point pro cess, it turns out that the pole lo cation is determined so that the tau function τ θ 2 always givessquare ro otto a w ell-normalized (gap-)probability distribution function. This concludes the proof of this prop osition. □ Up to notations or conv entions that v ary widely (cf. the proof of Proposition C for a discussion on this p oin t and related matters in the general four-parameter case of the Ok amoto–Jimbo-Miwa sigma form), all what precedes is already presen t and essen tially w ell kno wn in the classical or mo dern literature ab out the Bonnet P VI solution of the mean curv ature ODE. The most demanding issue is Proposition ( B.2 ), which addresses the P VI solution for the metric factor of the co- dimension one Bonnet P VI family . Its existence relies on very sp eciﬁc prop erties of Painlev ´ e functions that are little kno wn, and even less commonly used. It is also a crucial step for our determination of the p ersistence prob- abilit y distribution function, in particular to assess the genuine tr ansc endenc e of the corresp onding P VI . Pr o of (of the Proposition B.2 ) Our starting p oin t is the equation satisﬁed by the temp oral deriv ative of the explicitly time-dep enden t Hamiltonian ev aluated on the equations of motion. F ollowing [ 93 ] (p. 353), the computation b egins by recalling a simple but fundamental fact: using the chain-rule, one has d d s H VI ( p ( s ) , q ( s ) , s ) =     ∂ ∂ s H VI ( p, q , s ) + p ′ ∂ ∂ p H VI ( p, q , s ) + q ′ ∂ ∂ p H VI ( p, q , s ) | {z } 0     p = p ( s ) ,q = q ( s ) (B48) (where p ′ = d p d s and q ′ = d q d s ), since the last t wo terms cancel with eac h other b y the v ery deﬁnition of Hamilton’s equations. Hence the ov erall time-dependence of the temp oral deriv ative of the Hamiltonian ev aluated on the equations of motion can only originate from the built-in, explicit temp oral dep endence ∂ H VI ∂ s , which b y Ok amoto’s construction is itself rigidly determined by demanding its isomonodromy . T aking also accoun t in the deﬁnition ( B42 ) of the auxiliary Hamiltonian function h the pieces coming from the ov erall factor s ( s − 1) and from the aﬃne shift (cf. ( C6 )–( C7 )), one thereb y obtains for arbitrary mono drom y exp onents  θ α , θ β , θ γ , θ δ  d h d s = −  p 2 − p  θ β q + θ γ q − 1  q ( q − 1) − ( θ β + θ γ ) 2 4 , (B49) (matc hing (2.2) in [ 93 ], or (2.8) in [ 51 ], since the latter authors use the notations b 1 = θ β + θ γ 2 , b 2 = θ β − θ γ 2 ), where q = q ( s ) is a P VI function with arbitrary co eﬃcients [ α, β , γ , δ ] = 36  θ 2 α 2 , − θ 2 β 2 , θ 2 γ 2 , 1 − θ 2 δ 2  , and p = p ( s ) = p { θ β ,θ γ ,θ δ } is the conjugate impulsion determined through ( B43 ). Let us no w specialize ( B49 ) for the particular case of a P VI function and associated im- pulsion with a set of codimension-tw o monodromy exponents where (at least) t wo of them are zero. Without loss of generalit y (up to a fractional linear transformation in the indep enden t v ariable if necessary), we c ho ose to parametrize them as { 2 θ α , 0 , 0 , 2 θ δ } ( p,q ,s ) . (B50) Adding subscripts to the notation w e hav e used so far pro vides a simple means of referring to the impulsion, the position, and the independent v ariable of the v arious P VI that will show up. The arbitrary factors 2 that we hav e also ascribed in ( B50 ) for the monodromy exponents at inﬁnity and around the indep enden t v ariable will also simplify some of the subsequent expressions. An y P VI ha ving a set of mono drom y exp onents such as ( B50 ) where tw o of them are explicitly zero is amenable to a quadratic folding transformation. Applying the formulas recalled in App endix C , in particular Prop osition C.1 there, one ﬁnds by direct algebraic substitution, i.e. without an y diﬀerential elimination, that the right-hand-side of ( B 49 )    ( B 50 ) = − p 2 q ( q − 1) = −  2 P Q −  θ α + θ δ − 1 2  2 , (B51) with p = p ( s ), q = q ( s ), and where P = P ( t ) and Q = Q ( t ) are the resp ectiv e impulsion and p osition for another P VI with indep enden t v ariable t deﬁned through s = 1 2 + 1 4  √ t + 1 √ t  , (B52) and tw o couples of pairwise equal mono drom y exp onents { θ α , θ α , θ δ , θ δ } ( P,Q,t ) , (B53) whic h are therefore “split up” and “half-shared” compared to the set ( B50 ). The square ro ots in ( B52 ) en tail that there is algebraic branching, with tw o possible solutions for the indep enden t v ariable t of Q ( t ), the folded P VI . W e conv eniently denote these t wo solutions inv erse from eac h other as t ± ( s ), so that t = t ± 1 ( s ) =  2 s − 1 ± 2 p s ( s − 1)  2 . (B54) Ha ving in mind to apply for the real analytic Bonnet surfaces this quadratic P VI folding transformation, a v ery con venien t wa y to take in to accoun t the tw o p ossible signs abov e is to restrict ourselves to the independent v ariable parametrization ( B36 ) for s = s ε ( ν Re z ), since this amounts to write for t = t ( ν Re z ) (or equiv alently its inv erse) t = t ( ν Re z ) = coth 2  ν Re z 2  or tanh 2  ν Re z 2  . (B55) Of course, the relation ( B52 ) is more generally v alid for s, t ∈ C , since it is a simple transform of the famous Jouk owski conformal mapping z  → w = w ( z ) = 1 2  z + 1 z  . T o justify ( B55 ), note that whatev er b e the sign c hosen for ν = ± √ ν 2 (in case B sa y) and the c hoice of region of Re z (determined b y the Hopf factor ( B12 )), the t wo formulae ( B52 ) and ( B55 ) are alwa ys equiv alent because of the coth-duplication form ula (or cot if ν 2 < 0 in case A) 2 coth ( ν Re z ) = coth  ν Re z 2  + tanh  ν Re z 2  . (B56) 37 As for the tw o possible c hoices for the v ariable t = t ± 1 ( ν Re z ), they come from inv erting the square ro ot in ( B52 ) through ( B54 ). That sign is independent from ε , the one whic h keeps trac k in ( B23 ) of the orientation of the underlying surface, since the expressions in ( B55 ) are even in ν (and Re z ), hence unchanged if ν  → εν (and/or Re z  → − Re z ). Henceforth w e shall alwa ys assume that the square ro ot is determined by contin uity from the origin on the p ositiv e real axis, demanding that p t − 1 ( ν Re z ) = tanh ( ν Re z / 2) > 0 when ν 2 > 0 and ν Re z > 0. The crucial point no w is to observe that some signiﬁcant simpliﬁcation tak es place when the co-dimension t wo set of monodromy exponents ( B50 ) and ( B53 ) for these folded P VI ob ey the additional condition 2( θ α + θ δ ) − 1 = 0 , (B57) and that this relation can b e fulﬁlled for the reduced mean curv ature function h ( s ) = h θ 2 ( s ) of the Bonnet surfaces family parametrized by (the square of ) a single mono drom y exp onen t. Before w e embark on detailing the algebra, let us make a more conceptual remark ab out the meaning of ( B57 ), whic h will b e also decisiv e to prov e the gen uine transcendence of the P VI o ccurring in the p ersistence problem. What happens is that not only the relation ( B57 ) transforms ( B51 ) in to the m uch simpler h ′ = − (2 P Q ) 2 , but ab o ve all it is the ﬁngerprint of the deep algebraic structure at the heart of the P VI symmetries, namely the aﬃne D 4 W eyl ro ot system. Indeed, the constraint ( B57 ) corresp onds to sit on the chamb er of the wal l Θ = 0 of this ro ot system, since for generic mono drom y exp onen ts this parameter Θ is deﬁned through the aﬃne relationship θ α + θ β + 2 Θ + θ γ + θ δ = 1 . (B58) W e recall (see App endix C , or e.g. [ 51 ] for more details) that reﬂections in the abov e aﬃne theta -parameter space are realized as birational transformations betw een solutions, precisely with the p olynomial Hamiltonian constructed by Ok amoto to uncov er this D 4 symmetry . In particular, from a given “seed” P VI solution Q = Q ( t ) with monodromy exp onen ts  θ j  j suc h that Θ  = 0, the elementary Ok amoto transformation generates through Q  → Q ± Θ/P another P VI (actually , taking care of all possible signs, up to 2 × 2 4 = 32 contiguous ones), with shifted mono dromy exponents  θ j ± Θ  j . Con versely , the condition Θ = 0 is ne c essary — but not suﬃcient — to giv e rise to the so- called classic al solutions of P VI , which are either Riccati-transformed of the hypergeometric function 2 F 1 (as the tw o-parameter family encountered in [ 51 ]), or algebraic ones (such as t 7→ √ t ). In practice, hypergeometric solutions can b e obtained by demanding that in the Hamiltonian formalism their impulsion b e P ≡ 0. Since the metric function for Bonnet surfaces is directly proportional to the impulsion of the asso ciated folded P VI , cf. ( B62 ), the condition P = 0 has therefore to b e scrutinized in detail. It happ ens that for the Bonnet surfaces encountered in this work and coming from a determinan tal p oin t pro cess generated b y the probability conv olution kernels ( 2.38 ), b oth classical, hypergeometric P VI solutions and transcendental ones are realized, and that the Bonnet-P VI (of t yp e B) occurring in the p ersistence problem is genuinely transcenden tal, with P = P ( t )  = 0. F or the moment, it remains to chec k that one can ﬁnd a set of (signed) mono drom y exp onen ts holding for the Bonnet-P VI mean curv ature function which also satisﬁes the co dimension-one constrain t ( B57 ). This is indeed the case for the set i), which recalling ( B27 ) corresp onds to θ α = ε α θ , θ δ = 1 + ε δ θ (and θ β = θ γ = 0) if one c ho oses opp osite relative signs ε α ε δ = − 1 for the mono dromy exp onents at inﬁnit y and around the independent v ari- able. The sole resulting sign can therefore be ascrib ed through θ , and this gives rise to a pair of sets having suitable mono drom y exponents for our purp ose, to wit  θ α , θ β , θ γ , θ δ  = { ε θ θ , 0 , 0 , 1 − ε θ θ } ( p,q ,s ) , ε 2 θ = 1 . (B59) 38 Indeed, such a choice entails that the s -deriv ative of the reduced mean curv ature h = h θ 2 ( s ) — the latter function being b y construction the same either for set i) or ii) — is also directly related through a quadratic transformation to another P VI with impulsion P ( t ) and p osition Q ( t ) according to d h θ 2 ( s ) d s    s = s ε ( ν Re z ) = − 4 P 2 ( t ) Q 2 ( t )    t = t ± 1 (Re z ) . (B60) This P VI has therefore tw o couples of pairwise equal monodromy exp onen ts θ α = θ β and θ γ = θ δ solely parametrized by θ  θ α , θ β , θ γ , θ δ  =  θ 2 , θ 2 , 1 − θ 2 , 1 − θ 2  ( P,Q,t ) , (B61) or, and equiv alently due to ( B59 ), the set obtained by changing ev erywhere ab o ve θ  → − θ . No w w e combine our deﬁnition ( B31 ) of the reduced metric function g = g θ 2 ( t ) for a Bonnet surface, along with the results which on the one hand express G 2 = G 2 (Re z ) diﬀeren tially in terms of the reduced mean curv ature h = h θ 2 ( s ), and on the other hand that same quantit y algebraically in terms of the impulsion and the p osition of the folded P VI with indep enden t v ariable t , in order to arrive at our fundamental relationship ( B35 ) G 2 (Re z ) ( B 25 ) = d h θ 2 ( s ) d s    s = s ε ( ν Re z ) ( B 31 ) = − g 2 θ 2 ( t )    t = t ± 1 ( ν Re z ) ( B 60 ) = − 4 P 2 ( t ) Q 2 ( t )    t = t ± 1 ( ν Re z ) . (B62) Hence for short g = ± √ − h ′ = ± p 4 P 2 Q 2 , so that we obtain (choosing say + signs everywhere when taking the square roots) g θ 2 ( t ) = 2 P ( t ) Q ( t ) = ( t − 1) Q ( Q − 1)( Q − t )  t Q d Q d t − 1 2 − θ 2( t − 1)  Q − t Q  , (B63) where the last equality comes from ( B44 ), after expressing for the set of monodromy exp onen ts ( B61 ) the corresponding P = P ( t ) in terms of Q = Q ( t ) and of its deriv ative d Q/ d t , along with some algebraic rearrangemen t. A somewhat more compact formula — the one given as ( B33 ) in the lemma — can b e arriv ed at if one uses the permutation symmetry (denoted π 2 ( x ) in T able (3.3) p. 723 of [ 117 ], and r 1 in T able 1 p. 43 of [ 51 ]) which exchanges θ α ↔ θ δ and θ β ↔ θ γ while preserving the indep enden t v ariable in Ok amoto’s Hamiltonian according to the birational transformation ( P , Q, t ) { θ α ,θ β ,θ γ ,θ δ }  →  e P = − P ( Q − t ) 2 − Θ ( Q − t ) t ( t − 1) , e Q = ( Q − 1) t Q − t , e t = t  { θ δ ,θ γ ,θ β ,θ α } . (B64) W e apply this inv olution for the particular set ( B61 ) sitting on the c hamber Θ = 0 of the D 4 ro ot system. The condition ( B57 ) is (of course) preserv ed under the permutation θ α ↔ θ δ , and one obtains g θ 2 ( t ) = − 2 e P  e Q − 1   e Q − 1  t − 1 = − " t e Q d e Q d t − 1 2 − 1 − θ 2( t − 1)  e Q − t e Q  # , (B65) whic h is ( B33 ) as claimed (up to the notational tildes and the irrelev ant global sign), the ﬁnal expression b et ween brack ets being also the same as ( B63 ) after exchanging θ ↔ 1 − θ , since for this permutation 2 θ α ↔ 2 θ δ . Let us also remark en p assant , and as already hin ted at after ( B58 ), the existence of the elemen tary (zero-parameter) P VI algebraic solution Q ( t ) = ± √ t , which a priori is alwa ys allo wed here with a couple of pairwise equal monodromy exponents such as ( B53 ), whatev er their v alues (cf. [ 34 ], equation (B.193), p. 344). Y et, suc h an admissible solution of the P VI 39 equation would automatically mak e the reduced metric function iden tically zero, g θ 2 ( t ) ≡ 0, b y separately “annihilating” in a diﬀerential or algebraic fashion (and whatever θ ) the ﬁrst t wo (= t Q d Q d t − 1 2 ) and the last t wo terms (= Q − t Q ) appearing within the brack ets of ( B63 ) (and similarly for ( B65 )). F urthermore, this algebraic solution, as all the others of this nature, has zero-parameter left to accommo date for the initial conditions, and it has therefore to b e discarded here, since it cannot take into account the thinning parameter o ccurring in our k ernels. Finally , under the change of the independent and dep enden t v ariables (Re z , G 2 )  → ( t, − g 2 ) (for either c hoice of t = t ± 1 ( ν Re z )), one can c heck that the ODE ( B21 ) for the Bon- net metric is transformed in to the following one-parameter instance of Chazy C VI equation in rational co ordinates, viz. 1 2  d 2 g d t 2 +  1 2 t + 1 t − 1  d g d t + 2 g 3 − θ 2 g t ( t − 1) 2  2 = 2  1 2 t − 1 t − 1  2 g 2 "  d g d t  2 + g 4 − θ 2 g 2 t ( t − 1) 2 # . (B66) In this one-parameter situation, one is thereb y led to the equation ( C20 ). Namely , plug- ging in ( B66 ) either expression ( B63 ) or ( B65 ) for g = ± g θ 2 ( t ) (the sign b eing irrelev ant, since ( B66 ) is even in g ), and using the P VI equation for Q ( t ) (and corresp ondingly for e Q ( t )) to express the second and third-order deriv ativ es d 2 Q/ d t 2 , d 3 Q/ d t 3 in terms of the pow ers of d Q/ d t , one obtains a p olynomial in d Q/ d t (or d e Q/ d t ), of degree six (and without constant term), which has to be identically zero. By iden tiﬁcation of its resulting (huge!) coeﬃcients, the scrupulous reader can verify that ( B66 ) is indeed iden tically satisﬁed when the mon- o drom y exp onen ts of the corresp onding P VI ’s hav e resp ectiv e v alues ( B61 ) or ( B32 ). Note ﬁnally that ( B66 ) is even in θ , even though the form ulas ( B63 ) or ( B65 ) break explicitly the parit y in the mono drom y exp onen t. This terminates the long pro of of this Proposition B.2 and concludes this App endix. □ C Three remark able second order nonlinear ODEs: P VI , H VI , C VI This app endix summarizes the links b et ween three remark able second order nonlinear ODES, and recalls the folding transformation of P VI . The ﬁrst of these ODEs, the sixth Painlev ´ e equation P VI w ′′ = 1 2  1 w + 1 w − 1 + 1 w − z  w ′ 2 −  1 z + 1 z − 1 + 1 w − z  w ′ (C1) + w ( w − 1)( w − z ) z 2 ( z − 1) 2  α + β z w 2 + γ z − 1 ( w − 1) 2 + δ z ( z − 1) ( w − z ) 2  , has for general solution a function, by construction also called P VI , which is transcenden tal for generic v alues of its four “monodromy exponents” θ j ,  θ 2 α , θ 2 β , θ 2 γ , θ 2 δ  = ( 2 α, − 2 β , 2 γ , 1 − 2 δ ) , (C2) i.e. not expressible in terms of solutions of ODEs of order one or linearizable. The second ODE is ob eyed by a Hamiltonian of P VI [ 80 ], whose position v ariable q is equal to the w of P VI , suitably normalized so as to obey a v ery symmetric second 40 degree ODE [ 26 ], the so-called Ok amoto–Jim b o-Miw a sigma-form of P VI [ 91 ] [ 67 ], h ′ ( s ( s − 1) h ′′ ) 2 +  h ′ 2 − 2 h ′ ( sh ′ − h ) + b 1 b 2 b 3 b 4  2 − 4 Y j =1 ( h ′ + b 2 j ) = 0 , (C3) 2 b 1 = θ β + θ γ , 2 b 2 = θ β − θ γ , 2 b 3 = θ δ − 1 + θ α , 2 b 4 = θ δ − 1 − θ α . (C4) (W e hope that no confusion arises with the p osition q in the Hamiltonian formalism, and the notation used in the Introduction for the n umber of states of the Potts mo del.) This Hamiltonian H VI ( q , p, s ) is an aﬃne transform of h deﬁned by h ( s ) : = s ( s − 1) H VI ( q ( s ) , p ( s ) , s ) + As + B , (C5) 4 A : = − θ 2 α + ( θ δ − 1) 2 + 2( θ β + θ γ )( θ δ − 1) ≡ 4( b 1 b 3 + b 1 b 4 + b 3 b 4 ) , (C6) 8 B : = θ 2 α − θ 2 β + θ 2 γ − ( θ δ − 1) 2 − 4 θ β ( θ δ − 1) ≡ − 4 Y 1 ≤ j 0 . (D1) What the Borodin-Ok ounko v formula achiev es is to provide us with a global expres- sion v alid for all  > 0 of the F redholm determinant D θ (  ; ξ ), hence of its  -logarithmic deriv ative d d  log D θ (  ; ξ ) ≡ H θ (  ; ξ ) ,  > 0 . This prov es in particular the existence of and the (ﬁnite and negative) v alue for the asymptotic mean curv ature of this family of Bonnet-P VI surfaces lim ℓ → + ∞ H θ (  ; ξ ) = 1 2 θ 2 −  2 π arccos h p 1 − ξ cos ( π θ/ 2 ) i  2 ! , 0 ≤ θ 2 < 1 , 0 < ξ < 1 . (D2) Notice that the sp ecialization of ( D2 ) to θ = 1 / 2 and ξ = ξ ( m ) = 1 − m 2 giv es back the expression ( 1.4 ) for the persistence exp onen t. The unique negativ e solution to ( D1 ) H θ ( · ; ξ ) decreasing on R + with a ﬁnite limit at inﬁnit y is therefore globally deﬁned, and each mem b er of this t wo-parameter family ma y be view ed as the analog of the Hastings and McLeo d solution of P II [ 61 ] appearing in all the T racy-Widom distributions. Y et, a rather subtle but crucial p oint that we shall elab orate on at length b elo w is that for all v alues of θ 2 < 1 the equation ( D2 ) also remains v alid in the limiting case ξ → 1 − (and thus in particular in the m = 0 symmetric p ersistence Ising case), this essentially for probabilistic reasons — in short, Soshniko v’s theorem [ 109 ]. This is even though the Wiener-Hopf factorization for the kernels K θ formally fails there, the corresponding sym b ol ha ving a so-called Fisher-Hartwig singularity . The setup required to implemen t the BO result will be achiev ed in several steps. • Switc h from the thinning parameter ξ to a uniformizing parameter ϕ deﬁned through ξ = cos ( π θ ) − cos ( π ϕ ) cos ( π θ ) + 1 , and suc h that 0 < θ 2 < ϕ 2 < 1 for 0 < ξ < 1 . (D3) 44 Note that conv ersely w e ha ve ϕ : = 2 π arccos h p 1 − ξ . cos ( π θ/ 2 ) i ≡ 2 π arcsin  q ξ + (1 − ξ ) · sin 2 ( π θ/ 2)  , (D4) these t wo expressions b eing equiv alent on the common branc h [0 , π 2 ] of the tw o recipro cal trigonometric functions if ϕ is c hosen p ositiv e (and < 1) in ( D3 ). F or an y 0 < ξ < 1, we shall also assume 0 < θ , so that we alwa ys hav e 0 < θ < ϕ < 1, without loss of generalit y since the k ernel K θ is even in θ . Most of our ﬁnal expressions turn out to b e enen functions of both θ and ϕ , or this will be easy to restore. As usual, the case θ = 0 is obtained without any problem by passing to the limit, this temp orary res triction making easier some intermediate contour integral manipulations. One of the adv antages of the bijective parametrization ( D4 ) is that the Wiener-Hopf symb ol c W θ,φ for ξ K θ , which is conv entionally deﬁned as the F ourier transform of Id − ξ K θ , has a v ery symmetric form in θ , ϕ and the F ourier indep enden t v ariable u : c W θ,φ ( u ) : = F [Id − ξ K θ ]( u ) = cosh ( π u ) + cos ( π ϕ ) cosh ( π u ) + cos ( π θ ) . (D5) The normalization for our direct and inv erse F ourier transforms is f ( x ) = Z R d u 2 π e − i xu b f ( u ) , b f ( u ) = Z R d x e i ux f ( x ) , (D6) with the resp ectiv e abbreviations f = F − 1 [ b f ] , b f = F [ f ]. As a means of understand- ing the origin of ( D3 ), w e recall in particular (and this is a tabulated cosine F ourier transform) that for the K θ k ernel b K θ ( u ) = 1 + cos ( π θ ) cosh ( π u ) + cos ( π θ ) , (D7) the v alue at the origin in F ourier space b K θ (0) = R R K θ = 1 ensuring the probabilit y normalization on the full real line of the point process generated b y this kernel. F rom now on, when ξ < 1, thus also ϕ < 1, we index with the subscripts θ , ϕ the quan tities of interest, so that the mean curv ature function of this tw o-parameter family of Bonnet surfaces is giv en for all  > 0 b y the logarithmic deriv ative H θ,φ (  ) = d d  log D θ,φ (  ) , 0 < θ < ϕ < 1 , (D8) iden tifying also D θ,φ (  ) ≡ D θ (  ; ξ ) thanks the bijectiv e parametrization ξ ↔ ϕ . Of course, by F ourier conjugation, the F redholm determinant of the “truncated” Wiener-Hopf op erator (as it customarily referred to in the corresp onding literature) with symbol c W θ,φ , and that for the thinned k ernel ξ K θ (restricted to [0 ,  ]), are the same. 45 A t this stage, the pro of must b e split in to t wo cases, ξ  = 1 and ξ = 1. Indeed, it is only for ϕ < 1 that the function u 7→ c W θ,φ ( u ) given by ( D5 ) never v anishes on the real axis and is b ounded (strictly) by one for all ﬁnite u , therefore ( D7 ) can be extended as a complex analytic function in the strip − (1 − θ ) < Im u < 1 − θ . If and only if ϕ < 1 (i.e. ξ < 1), the sym b ol c W θ,φ is therefore factorizable in the sense of Wiener and Hopf, and the factorization v alid in the ab o v e strip is just obtained b y insp ection using t wice the elementary cos(h)-addition relationship, then (four times!) the Gamma function complemen t form ula Γ( z )Γ(1 − z ) = π / sin ( π z ), c W θ,φ ( u ) = F θ,φ (i u/ 2) F θ,φ ( − i u/ 2) (D9) where F θ,φ ( z ) : = Γ( 1+ θ 2 − z )Γ( 1 − θ 2 − z ) Γ( 1+ φ 2 − z )Γ( 1 − φ 2 − z ) 0 < θ < ϕ < 1 . (D10) The t wo Wiener-Hopf factors, traditionally denoted as F ± , and whic h share the ab o v e-mentioned strip as a common region of analyticity , are thus F ± ( u ) = F θ,φ ( ± i u/ 2), where z 7→ F θ,φ ( z ) is meromorphic, with t wo sets of simple p oles and simple zeros on the positive real axis in the z v ariable, whic h interlace but never o verlap. Note also the nice symmetry for their in verses in the strip: 1 /F θ,φ (i u/ 2) = F φ,θ ( − i u/ 2). • Iﬀ ξ  = 1, application to the then factorizable Wiener-Hopf sym b ol of the explicit form ula of Borodin and Ok ounko v [ 17 ], ∀  > 0 : log D θ,φ (  ) = A θ,φ ·  + B θ,φ + log Det [ Id − L θ,φ ] ↾ [ ℓ, + ∞ ) (D11) the three terms inv olving F ourier in tegrals as no w detailed. The ﬁrst t wo terms of ( D11 ) are known explicitly in terms of the inv erse F ourier transform of the logarithm of the Wiener-Hopf symbol, e θ,φ ( x ) = F − 1 h log c W θ,φ i ( x ) , (D12) the corresponding Kac-Akhiezer form ulas [ 71 ] being giv en b y A θ,φ = lim x → 0 e θ,φ ( x ) , B θ,φ = Z ∞ 0 d x x · e θ,φ ( x ) · e θ,φ ( − x ) . (D13) In our con tin uous con volution op erator setting, these t wo terms are resp ectiv ely the analogs of the weak and strong forms of Szeg¨ o theorem for T oeplitz (discrete) determinan ts. The function e θ,φ (whic h here is even, lik e the k ernel K θ ) can b e computed using the residue theorem and a suitable contour, which even tually amoun ts to summing a geometric series. This is the metho d follow ed in the pro of of Ref. [ 50 , Corollary 6] for the p ersistence sec h kernel. A shortcut exists for the more general K θ k ernel, whic h consists in using the analyticity of the nev er v anishing sym b ol log c W θ,φ in the parameter ϕ : if one diﬀerentiates e θ,φ ( x ) with resp ect to ϕ , one recognizes in 46 the resulting in tegral merely the F ourier transform of the K φ k ernel (up to some prefactor) ∂ ∂ ϕ e θ,φ ( x ) = − π sin ( π ϕ ) Z R d w 2 π e − i xw cosh ( π w ) + cos ( π ϕ ) = − sinh ( ϕx ) sinh x . (D14) In tegrating back ( D14 ) from the ob vious limiting v alue lim φ → θ + e θ,φ ( x ) = e θ,θ ( x ) = 0 (corresponding to ξ → 0 + ) yields e θ,φ ( x ) = cosh ( θ x ) − cosh ( ϕx ) x sinh x . (D15) Consequen tly , this determines (b y con tinuit y of the F ourier transform) the ﬁrst term of the r.h.s. of ( D11 ), the so-called ge ometric me ans for our sym b ol A θ,φ : = lim ℓ → + ∞ log D θ,φ (  )  = lim x → 0 e θ,φ ( x ) = θ 2 − ϕ 2 2 . (D16) T o obtain the second term B θ,φ , we use its equiv alen t expression as an integral on the imaginary axis inv olving the logarithm of the t w o Wiener-Hopf factors B θ,φ = Z +i ∞ − i ∞ d z 2i π log F θ,φ ( − z ) d d z log F θ,φ ( z ) . (D17) (This is a known general formula that can b e obtained either b y a standard F ourier computation, see e.g. [ 6 ], or as a certain Sob olev space isometry [ 22 ].) Because of the meromorphy of F θ,φ , the ab o ve integral only inv olves simple p oles. F ollo wing exactly the metho d of pro of for Ref. [ 6 , Lem ma 3.26], a result for the sech kernel (and with their β = ϕ − 1 / 2 in our notations) which, as in triguingly remark ed b y Basor and Ehrhardt [ 6 ], seemed to b e already in 2005 “of in terest in its own”, w e even tually obtain for the limiting amplitude B θ,φ of the determinan t the very symmetric expression exp B θ,φ = G 2  1 + θ + φ 2  G 2  1 − θ + φ 2  G 2  1 + θ − φ 2  G 2  1 − θ − φ 2  G (1 + θ ) G (1 − θ ) G (1 + ϕ ) G (1 − ϕ ) , (D18) in which G is the Barnes function, essen tially deﬁned b y G ( z + 1) = Γ( z ) G ( z ), G (1) = 1. (W e hop e no confusion takes place with the notation used in App endix C for the solution of the Chazy C VI ODE ( C20 ) which gives the metric for Bonnet surfaces.) Note that for ϕ = 1 (corresp onding to ξ = 1), the Barnes function G (1 − φ ) in the denominator v anishes and, as a signal of the formal failure of the Wiener-Hopf factorization, the amplitude abov e therefore becomes inﬁnite. Of course, a ﬁnite amplitude exists, and we shall detail later ho w to obtain the correct, “regularized” result. T o compute the third term of of ( D11 ), w e need to introduce another function f θ,φ ( x ) deﬁned for x > 0 through the in verse F ourier transform of the ratio of the 47 t wo Wiener-Hopf factors F ± ( u ) = F θ,φ ( ± i u/ 2) f θ,φ ( x ) : = F − 1  F − F + − 1  ( x ) = F − 1  F + F − − 1  ( − x ) , (D19) the second equality holding due to the ev enness of the sym b ol. This function f θ,φ determines the operator L θ,φ app earing in the remainder term of the Boro din-Ok ounko v form ula ( D11 ) as the squar e of a Hankel op er ator , whose matrix elemen ts when that L θ,φ acts on L 2 ([ , + ∞ )) are ∀ x > 0 , ∀ y > 0 : L θ,φ ( x, y ) ↾ [ ℓ, + ∞ ] : = Z + ∞ ℓ d w f θ,φ ( x + w ) f θ,φ ( w + y ) . (D20) Note in particular the simple expression v alid for arbitrary  > 0 of the second deriv ative of the trace of this op erator, d 2 d  2 T r [ L θ,φ ] ↾ [ ℓ, + ∞ ] = ( f θ,φ (  ) ) 2 . (D21) It remains to compute in a suﬃciently explicit w a y f θ,φ . Using the θ ↔ ϕ symmetry 1 /F θ,φ ( z ) = F φ,θ ( − z ) of the tw o Wiener-Hopf factors, its deﬁnition ( D19 ) reduces to an in tegral ` a la Mel lin-Barnes inv olving a ratio of well-balanced Gamma functions f θ,φ ( x ) = 2 Z i ∞ − i ∞ d z 2i π e − 2 xz Γ( z + 1+ θ 2 )Γ( z + 1 − θ 2 )Γ( − z + 1+ φ 2 )Γ( − z + 1 − φ 2 ) Γ( z + 1+ φ 2 )Γ( z + 1 − φ 2 )Γ( − z + 1+ θ 2 )Γ( − z + 1 − θ 2 ) − 1 ! . (D22) Closing up the con tour in the righ t-half plane and picking up the residues of the t wo strings of p oles originating from the tw o top rightmost Gamma functions ab o ve one can express (as in [ 3 , 6 ]) f θ,φ as the sum of t wo c onver gent h yp ergeometric series 4 F 3 . F or simplicit y , we do not write the corresponding formula, since to determine the large  b eha vior of the remainder term in the Borodin-Okounk ov form ula ( D11 ), it is suﬃcient to keep the ﬁrst t wo terms in the series f θ,φ (  ) = θ 2 − ϕ 2 2 Γ  θ − φ 2  Γ  − θ − φ 2  Γ  θ + φ 2  Γ  − θ + φ 2  Γ( ϕ ) Γ(1 − ϕ ) e − (1 − φ ) ℓ + Γ  θ + φ 2  Γ  − θ + φ 2  Γ  θ − φ 2  Γ  − θ − φ 2  Γ( − ϕ ) Γ(1 + ϕ ) e − (1+ φ ) ℓ + . . . ! . (D23) When 0 < ϕ < 1, the second term is of course sub-dominan t for  ≫ 1, but we momen tarily display it in conjunction with the ﬁrst one to emphasize the ov erall parit y symmetry ϕ ↔ − ϕ exp ected from ( D3 ), while the omitted terms are all deca ying smaller than e − 2 ℓ (uniformly in 0 < ϕ 2 < 1). 48 Bearing this in mind, w e abbreviate the asymptotic form ( D23 ) as f θ,φ (  ) = c θ,φ e − (1 − φ ) ℓ + . . . , c θ,φ : = θ 2 − ϕ 2 2 Γ  θ − φ 2  Γ  − θ − φ 2  Γ  θ + φ 2  Γ  − θ + φ 2  Γ( ϕ ) Γ(1 − ϕ ) , (D24) where the sub-dominant corrections are at least O  e − (1 − φ ) ℓ  . W orking to this exp onen tially small low est order is suﬃcien t to determine the so- called Widom-Dyson c onstant C θ,φ o ccurring in the remainder term of the Boro din- Ok ounko v form ula log Det ( Id − L θ,φ ) ↾ [ ℓ, + ∞ ) = −C θ,φ e − 2(1 − φ ) ℓ + terms at least O  e − 2 ℓ  , (D25) with here C θ,φ =  c θ,φ 2(1 − ϕ )  2 =   ϕ 2 − θ 2 4 Γ( ϕ ) Γ(2 − ϕ ) Γ  θ − φ 2  Γ  − θ − φ 2  Γ  θ + φ 2  Γ  − θ + φ 2    2 . (D26) Indeed, the tail op erator L θ,φ ↾ ( ℓ, + ∞ ) is “very small”, meaning that due to ( D24 ) one can chec k that T r  L n θ,φ  ↾ ( ℓ, + ∞ ) ∼  T r [ L θ,φ ] ↾ ( ℓ, + ∞ )  n ,  ≫ 1 , (D27) with the single trace itself exponentially small T r [ L θ,φ ] ↾ ( ℓ, + ∞ ) = Z + ∞ 0 d x 1 Z ∞ 0 d x 2 ( f θ,φ (  + x 1 + x 2 ) ) 2 ∼  c θ,φ e − (1 − φ ) ℓ 2(1 − ϕ )  2 ,  ≫ 1 . (D28) This implies that the standard trace-log expansion of the determinan t can b e re- summed exactly in this asymptotic regime, b ecause then log Det [ Id − L θ,φ ] ↾ ( ℓ, + ∞ ) = − X n ≥ 1 1 n T r  L n θ,φ  ↾ ( ℓ, + ∞ ) ∼ − X n ≥ 1 1 n  T r [ L θ,φ ] ↾ ( ℓ, + ∞ )  n (D29) and therefore Det ( Id − L θ,φ ) ↾ [ ℓ, + ∞ )] ∼ e − T r [ L θ,φ ] ↾ [ ℓ, + ∞ )] = 1 − T r [ L θ,φ ] ↾ [ ℓ, + ∞ )] + · · · , (D30) up to terms deca ying exp onen tially faster than the single trace, which by using ( D28 ) th us pro ves ( D25 ) with the v alue ( D26 ). Putting all the pieces together, we therefore ha ve in the large  limit the follo wing expansion for the logarithm of D θ,φ (  ), the three constants b eing fully explicit and giv en resp ectiv ely by ( D16 ), ( D18 ), and ( D26 ) in terms of the parameters 0 ≤ θ < 49 ϕ < 1 (the limit θ → 0 nev er posing any problems), log D θ,φ (  ) = A θ,φ ·  + B θ,φ − C θ,φ e − 2(1 − φ ) ℓ + O  e − 2 ℓ  . (D31) Under the stated conditions on the parameters, this uniformly con vergen t expan- sion is also differentiable with resp ect to  . Hence the deﬁning Eq. ( D8 ) gives the expression for the mean curv ature function in the large  limit as H θ,φ (  ) = A θ,φ + 2(1 − ϕ ) C θ,φ e − 2(1 − φ ) ℓ + O  e − 2 ℓ  , (D32) and th us the asymptotic mean curv ature of these Bonnet surfaces at their sole um bilic p oin t: lim ℓ → + ∞ H θ,φ (  ) = A θ,φ = θ 2 − ϕ 2 2 , (D33) a negativ e v alue approac hed exp onen tially fast from ab o ve by ( D32 ). • When ξ = 1 (i.e. ϕ = 1), the Wiener-Hopf symbol has a so-called Fisher-Hartwig singularit y [ 49 ], because by ( D7 ) it v anishes at u = 0 with a double zero c W θ, 1 ( u ) = 1 − c K θ ( u ) ∼  π 2 cos ( π θ / 2)  2 u 2 . (D34) Since the logarithm of this sym b ol remains in tegrable, its geometric means is simply obtained b y con tin uity from the v alue at the origin of its in verse F ourier transform e θ, 1 ( x ), hence from ( D15 ) and ( D16 ) as a θ : = lim φ → 1 A θ,φ = θ 2 − 1 2 . (D35) Ho wev er, the Wiener-Hopf factorization ( D10 ) now breaks down, since eac h of the corresp onding factors F ± ( u ) = F θ, 1 ( ± i u/ 2) has a zero at u = 0, whic h en tails that the BO representation form ula ( D11 ) do es not exist anymore as it stands, as already testiﬁed b y the div ergence of the amplitude B θ,φ ( D18 ) in the ϕ → 1 limit. F ortunately , there exists a result by Widom, who prov es [ 119 , Theorem 4] that for a symbol with the simplest p ossible Fisher-Hartwig singularit y such as ( D34 ), the follo wing rigorous expansion holds for its F redholm determinan t log D θ (  ; ξ )    ξ =1 = a θ ·  + log (  + c θ ) + b θ + O  e − (1 − θ ) ℓ  ,  ≫ 1 , (D36) with explicitly kno wn constan ts b θ and c θ , the exponential error term b eing the best p ossible, since it comes from the existence of an exp onential moment R R d u e pu b K θ ( u ) of order p < 1 − θ , a tec hnical condition fulﬁlled here thanks to ( D7 ). In particular, ( D36 ) localizes exactly — i.e. with an error term strictly zero — for the pure Fisher- Hart wig symbol u 2 / ( u 2 + 1) corresp onding to the Mark ovian kernel K ( x − y ) = e −| x − y | , and for which the truncated F redholm determinant is equal to e − ℓ (1 + / 2) [ 87 ], one of the rare cases known for all  > 0. 50 The abov e w ork b y Widom is v ery little cited. F or the determinan t of a Wiener-Hopf sym b ol with a Fisher-Hartwig singularit y such as ( D34 ) and its generalizations, one rather ﬁnds in the literature (see, e.g., [ 7 , App endix A], and references therein) a conjectural expansion under the deprecated form log D θ (  ; ξ )    ξ =1 = a θ ·  + log  + b θ + O   − 1  ,  ≫ 1 . (D37) This corresp onds to replace formally for ϕ = 1 in the original BO form ula ( D31 ) the div erging B θ,φ  → log  + b θ for  large. Of course, ( D37 ) is a genuine mathematical consequence of ( D36 ) (and a muc h weak er one concerning the error term) if one expands the logarithmic piece log (  + c θ ) = log  + O (  − 1 ), with here the ﬁnite and computable constan t c θ 2 = d d z  F θ, 1 ( z ) F θ, 1 ( − z )     z =0 = Ψ  1 + θ 2  − Ψ  1 − θ 2  − 2Ψ(1) , (D38) (the common zero at the origin of the tw o Wiener-Hopf factors simplifying in the ratio), where Ψ( z ) = (Γ ′ / Γ)( z ) is the digamma function and Ψ(1) = − γ E Euler’s constan t. The regularized amplitude is giv en b y the no w con vergen t in tegral 1 exp ( b θ ) = exp  Z ∞ 0 d x x  ( e θ, 1 ( x ) ) 2 − 1 − e − x x  = π 2 4 θ 2 G 8 (1 / 2) G (1 − θ ) G (1 + θ ) G 4 (1 − θ / 2) G 4 (1 + θ / 2) , (D39) whic h is a one-parameter extrapolation of the v alue that Fitzgerald, T rib e, and Zab oronsky ha ve obtained in [ 50 ] by a direct study of the Pfaﬃan F redholm de- terminan t for the sec h k ernel. (The ﬁnal explicit expression giv en in ( D39 ) relies on some Barnes function duplication identities.) In the case relev an t for persistence ( θ = 1 / 2), we hav e found that ( D39 ) ev aluates in terms of the Glaisher-Kinkelin constan t A GK , that one also often encoun ters in Painlev ´ e connection problems, and whic h is related to the Riemann ζ function by log A GK = − ζ ′ ( − 1) + 1 / 12, so that here ev entually exp ( b θ )    θ =1 / 2 = e 6 ζ ′ ( − 1) 2 1 / 12 Γ(3 / 4) Γ(1 / 4) . (D40) Since the expansion ( D36 ) is uniformly conv ergent, w e thus obtain the asymptotic b eha vior of the Bonnet-P VI mean curv ature function asso ciated to the un thinned K θ k ernel: H θ (  ; ξ )    ξ =1 = θ 2 − 1 2 + 1  + O   − 2  ,  → + ∞ . (D41) All the results when ξ < 1 in this App endix can b e summarized in the Prop osition D.1. F or any 0 < θ < ϕ < 1 , wher e ϕ = ϕ θ ( ξ ) is the uniformizing p a- r ameter deﬁne d in ( D3 ), and bije ctively e quivalent to ξ = ξ θ ( ϕ ) , the Bor o din-Okounkov 1 W e thank G. Korchemsky for pointing [ 7 ] to us, and for showing us another method to obtain ( D39 ) from ( D18 ). 51 formula applie d to the Wiener-Hopf symb ol in F ourier sp ac e c W θ,φ ( u ) : = F [ Id − ξ θ ( ϕ ) K θ ] ( u ) = cosh ( π u ) + cos ( π ϕ ) cosh ( π u ) + cos ( π θ ) , − (1 − θ ) < Im u < (1 − θ ) , (D42) pr ovides for t he determinant D θ,φ (  ) of the trunc ate d op er ator c W θ,φ ↾ ℓ a r epr esentation valid for al l interval length  ∀  > 0 : log D θ,φ (  ) = A θ,φ ·  + B θ,φ + log Det [ Id − L θ,φ ] ↾ [ ℓ, + ∞ ) . (D43) • The Sze g¨ o-Kac-A khiezer formula gives its dominant exp onential b ehavior as  → + ∞ A θ,φ : = lim ℓ → + ∞ log D θ,φ (  )  = θ 2 − ϕ 2 2 , (D44) this (always ne gative) ge ometric me ans b eing c ontinuous when ϕ ↗ 1 (or ξ ↗ 1 e quivalently). • The suble ading term B θ,φ in ( D11 ) determines the asymptotic amplitude of the determinant, and t his pr efactor evaluates in terms of Barnes’ G -function to exp B θ,φ = G 2  1 + θ + φ 2  G 2  1 − θ + φ 2  G 2  1 + θ − φ 2  G 2  1 − θ − φ 2  G (1 + θ ) G (1 − θ ) G (1 + ϕ ) G (1 − ϕ ) . (D45) • The r emainder term involves the squar e of a b ounde d Hankel op er ator L θ,φ acting on the c omplementary interval [ , + ∞ ) , whose c omplic ate d but explicit expr ession is determine d by the Wiener-Hopf factors of the symb ol W θ,φ . 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