Gap probabilities in tiling models and Painlevé equations

Transcription

1 Gap probabilities in tiling models and Painlevé equations Alisa Knizel Columbia University Painlevé Equations and Applications: A Workshop in Memory of A. A. Kapaev August 26, / 17

2 Tiling model Lozenge tilings of a hexagon can be viewed as stepped surfaces. 2 / 17

3 Tiling model Consider the probability measure on the set of tilings defined by PpT q ωpt q Zpa, b, cq, where ωpt q ź ωp q. PT 3 / 17

4 Tiling model Consider the probability measure on the set of tilings defined by PpT q ωpt q Zpa, b, cq, where ωpt q ź ωp q. PT If we set ωp q 1 we will obtain the uniform measure. 3 / 17

5 Tiling model Consider the probability measure on the set of tilings defined by PpT q ωpt q Zpa, b, cq, where ωpt q ź ωp q. PT If we set ωp q 1 we will obtain the uniform measure. Let j be the coordinate of. Set ωp q q j, 1 ą q ą 0. 3 / 17

6 Tiling model Consider the probability measure on the set of tilings defined by PpT q ωpt q Zpa, b, cq, where ωpt q ź ωp q. PT If we set ωp q 1 we will obtain the uniform measure. Let j be the coordinate of. Set ωp q q j, 1 ą q ą 0. ωpt q constpa, b, cq q volume. 3 / 17

7 Tiling model Affine transformation 4 / 17

8 Tiling model Affine transformation 4 / 17

9 Tiling model Affine transformation establishes a bijection between tilings and non-intersecting paths: 5 / 17

10 Gap probability Fix a section t. Let the coordinates of the nodes be Cptq px 1,..., x N q. 6 / 17

11 Gap probability Fix a section t. Let the coordinates of the nodes be Cptq px 1,..., x N q. Definition The one-interval gap probability function D N s is D N s Probrx i s, s ` 1,..., Ms Probrmaxtx i u ă ss. 6 / 17

12 Orthogonal polynomial ensembles Let X be a discrete subset of R. Let ω be a positive-valued function on X with finite moments: ÿ x n ωpxq ă 8, n 0,... xpx 7 / 17

13 Orthogonal polynomial ensembles Let X be a discrete subset of R. Let ω be a positive-valued function on X with finite moments: ÿ x n ωpxq ă 8, n 0,... xpx Fix cardpxq ą k ą 0. The orthogonal polynomial ensemble on X is a probability measure on the set of all k-subsets of X given by Ppx 1,..., x k q 1 Z ź kź px i x j q 2 ωpx i q, 1ďiăjďk i 1 where Z is the normalizing constant. 7 / 17

14 q-hahn Theorem (Borodin, Gorin, Rains 2009) ProbtCptq px 1,..., x N qu const ź 0ďiăjďM pq xi q x j q 2 where ω t pxq is the weight function of the q-hahn polynomial ensemble up to a factor not depending on x. Nź ω t px i q, i 1 8 / 17

15 q-hahn weight Let M P Z ą0, 0 ă α ă q 1 and 0 ă β ă q 1 ω q Hahn pαq, pxq pαβqq x q M ; qq x pq, β 1q M, where ; qq x py 1,..., y i ; qq k py 1 ; qq k py i ; qq k, and py; qq k p1 yq p1 yq k 1 q. 9 / 17

16 q-hahn weight Let M P Z ą0, 0 ă α ă q 1 and 0 ă β ă q 1 ω q Hahn pαq, pxq pαβqq x q M ; qq x pq, β 1q M, where ; qq x py 1,..., y i ; qq k py 1 ; qq k py i ; qq k, and py; qq k p1 yq p1 yq k 1 q. In our case, for example S 1 ă t ă T S ` 1 M S ` N 1 α t N β t T N t ă S and T t S ą 0 M t ` N 1 α S N β S T N 9 / 17

17 Theorem [K. 16], q-volume case The gap probability Ds N for the q-hahn ensemble can be computed recursively D N s pdn s 2 q2 D N s 1 pr s 1 w qva 1 a 2 qpr s w qua 1 a 2 qpt s 1 qa 1 qpt s 1 qa 2 q, uva 1 a 2 pqa 1 a 3 qpqa 1 a 5 qpqa 2 a 4 qpqa 2 a 6 q 10 / 17

18 Theorem [K. 16], q-volume case The gap probability Ds N for the q-hahn ensemble can be computed recursively D N s pdn s 2 q2 D N s 1 pr s 1 w qva 1 a 2 qpr s w qua 1 a 2 qpt s 1 qa 1 qpt s 1 qa 2 q, uva 1 a 2 pqa 1 a 3 qpqa 1 a 5 qpqa 2 a 4 qpqa 2 a 6 q where the sequence pr s, t s q satisfies recursion pr s t s 1 ` 1qpr s 1 t s 1 ` 1q a 1 a 2 pt s 1 a 3 qpt s 1 a 4 qpt s 1 a 5 qpt s 1 a 6 q, a 3 a 4 a 5 a 6 pqt s 1 a 1 qpqt s 1 a 2 q pr s t s ` 1qpr s t s 1 ` 1q uvpa 1 a 2 q 2 pr s a 3 ` 1qpr s a 4 ` 1qpr s a 5 ` 1qpr s a 6 ` 1q. pr s w s va 1 a 2 qpqr s w s ua 1 a 2 q 10 / 17

19 Theorem [K. 16], q-volume case The gap probability Ds N for the q-hahn ensemble can be computed recursively D N s pdn s 2 q2 D N s 1 pr s 1 w qva 1 a 2 qpr s w qua 1 a 2 qpt s 1 qa 1 qpt s 1 qa 2 q, uva 1 a 2 pqa 1 a 3 qpqa 1 a 5 qpqa 2 a 4 qpqa 2 a 6 q where the sequence pr s, t s q satisfies recursion pr s t s 1 ` 1qpr s 1 t s 1 ` 1q a 1 a 2 pt s 1 a 3 qpt s 1 a 4 qpt s 1 a 5 qpt s 1 a 6 q, a 3 a 4 a 5 a 6 pqt s 1 a 1 qpqt s 1 a 2 q pr s t s ` 1qpr s t s 1 ` 1q uvpa 1 a 2 q 2 pr s a 3 ` 1qpr s a 4 ` 1qpr s a 5 ` 1qpr s a 6 ` 1q. pr s w s va 1 a 2 qpqr s w s ua 1 a 2 q The parameters u, v, w, a 1,..., a 6 and the initial conditions are explicitly computed in terms of α, β, q, s. 10 / 17

20 Limit shape [Cohn Larsen Propp 98], [Cohn Kenyon Propp 01], [Kenyon Okounkov 07]. 11 / 17

21 Limit shape [Cohn Larsen Propp 98], [Cohn Kenyon Propp 01], [Kenyon Okounkov 07]. Simulation by Leonid Petrov 11 / 17

22 Edge fluctuations Let all the sides linearly grow as M Ñ 8 and q Ñ 1. After the change of variables x c 1 M ` c 2 um 1 3 we present a graph of c 2 M 1 3 pd N px ` 1q D N pxqq, for q 0.99: M 2000 M / 17

23 Discrete RH Theorem [Borodin-Boyarchenko 02 ] Fix cardpxq ą k ą 0, and set wpψq 0 ωpψq 0 0 j. 13 / 17

24 Discrete RH Theorem [Borodin-Boyarchenko 02 ] Fix cardpxq ą k ą 0, and set wpψq 0 ωpψq 0 0 For any s ě k there exists unique analytic function m s pψq : CzN s Ñ MatpC, 2q with simple poles at points in N s such that Res m spψq lim m s pψqwpψq, x P N s ; ψ x ψñx j. 13 / 17

25 Discrete RH Theorem [Borodin-Boyarchenko 02 ] Fix cardpxq ą k ą 0, and set wpψq 0 ωpψq 0 0 For any s ě k there exists unique analytic function m s pψq : CzN s Ñ MatpC, 2q with simple poles at points in N s such that Res m spψq lim m s pψqwpψq, x P N s ; ψ x ψñx ψ k 0 m s pψq 0 ψ k where N s t0,..., s 1u. j I ` O ˆ 1 ψ as ψ Ñ 8, j. 13 / 17

26 Introduce matrix A s pzq m pq 1 s zqa pzqm 1 0 s pzq, «ff qωpx`1q where A 0 pzq ωpxq 0 with z q x / 17

27 Introduce matrix A s pzq m pq 1 s zqa pzqm 1 0 s pzq, «ff qωpx`1q where A 0 pzq ωpxq 0 with z q x. 0 1 In the q-hahn case qω q Hahnpx ` 1q ω q Hahn pxq pz αqq pz q M q αβpz qq pz β 1q M q. 14 / 17

28 Claim 1: The gap probabilities Ds N can be computed in terms a s of the matrix elements of A s 11 a12 s j a21 s a22 s ; 15 / 17

29 Claim 1: The gap probabilities Ds N can be computed in terms a s of the matrix elements of A s 11 a12 s j a21 s a22 s ; Claim 2: For any s matrix element a12 s Denote it by t s and a22 s rt ss by p s. has a unique zero. 15 / 17

30 Claim 1: The gap probabilities Ds N can be computed in terms a s of the matrix elements of A s 11 a12 s j a21 s a22 s ; Claim 2: For any s matrix element a12 s Denote it by t s and a22 s rt ss by p s. has a unique zero. Claim 3: Evolution pt s, p s q Ñ pt s`1, p s`1 q has the form of a discrete Painlevé equation of type A p1q / 17

31 Structure of a generic A s pzq a11 pzq a Apzq 12 pzq a 21 pzq a 22 pzq j w 0, Ap0q 0 w degpa 11 q ď 3, degpa 12 q ď 2, degpa 21 q ď 2, degpa 22 q ď 3 j, 16 / 17

32 Structure of a generic A s pzq a11 pzq a Apzq 12 pzq a 21 pzq a 22 pzq j w 0, Ap0q 0 w degpa 11 q ď 3, degpa 12 q ď 2, degpa 21 q ď 2, degpa 22 q ď 3 j, detpapzqq uvpz a 1 qpz a 2 qpz a 3 qpz a 4 qpz a 5 qpz a 6 q. 16 / 17

33 Structure of a generic A s pzq a11 pzq a Apzq 12 pzq a 21 pzq a 22 pzq j w 0, Ap0q 0 w degpa 11 q ď 3, degpa 12 q ď 2, degpa 21 q ď 2, degpa 22 q ď 3 j, detpapzqq uvpz a 1 qpz a 2 qpz a 3 qpz a 4 qpz a 5 qpz a 6 q. We also require that detpapzqq uvz 6 ` Opz 5 q; trpapzqq pu ` vqz 3 ` Opz 2 q. 16 / 17

34 Evolution When A s pzq Ñ A s`1 pzq pa s 1, a s 2,..., a s 6, u s, v s, w s q Ñ pa s`1 1, a s`1 2,..., a s`1 6, u s`1, v s`1, w s`1 q 17 / 17

35 Evolution When A s pzq Ñ A s`1 pzq pa s 1, a s 2,..., a s 6, u s, v s, w s q Ñ pa s`1 1, a s`1 2,..., a s`1 6, u s`1, v s`1, w s`1 q with a s`1 1 qa s, a s`1 1 qa s, w s`1 qw s ; a s`1 i a s i for i 3,..., 6; u s`1 u s, v s`1 v s ; pt s, p s q Ñ pt s`1, p s`1 q is given by qppa p1q 2 q. 17 / 17