Log-gases on a quadratic lattice via discrete loop equations Alisa - PowerPoint PPT Presentation

Tiling model Log-gases on a quadratic lattice via discrete loop equations Alisa Knizel Columbia University Joint work with Evgeni Dimitrov April 11, 2019 1 / 35

Tiling model Log-gases on a quadratic lattice Fix S Ă R , N ą 0 and w : S Ñ R ě 0 , w p x q ě 0 . We consider a class of probability measures P N p S , w q on all N -point subsets of S of the form N p ℓ i ´ ℓ j q 2 ¨ ź ź P N p ℓ 1 , ℓ 2 , . . . , ℓ n q9 w p ℓ i q , 1 ď i ă j ď N i “ 1 where ℓ i P S for i “ 1 , . . . , N , where S “ t q ´ x ` uq x : 0 ď x ď M u with q P p 0 , 1 q , x , M P Z ě 0 and u P r 0 , 1 q . 2 / 35

Tiling model Tiling model Lozenge tilings of a hexagon can be viewed as stepped surfaces. 3 / 35

Tiling model Tiling model Consider the probability measure on the set of tilings defined by ω p T q ź P p T q “ Z p a , b , c q , where ω p T q “ ω p ✸ q . ✸ P T 4 / 35

Tiling model Tiling model Consider the probability measure on the set of tilings defined by ω p T q ź P p T q “ Z p a , b , c q , where ω p T q “ ω p ✸ q . ✸ P T • If we set ω p ✸ q “ 1 we will obtain the uniform measure . 4 / 35

Tiling model Tiling model Consider the probability measure on the set of tilings defined by ω p T q ź P p T q “ Z p a , b , c q , where ω p T q “ ω p ✸ q . ✸ P T • If we set ω p ✸ q “ 1 we will obtain the uniform measure . • Let j be the coordinate of ✸ . Set ω p ✸ q “ q ´ j , 1 ą q ą 0 . 4 / 35

Tiling model Tiling model Consider the probability measure on the set of tilings defined by ω p T q ź P p T q “ Z p a , b , c q , where ω p T q “ ω p ✸ q . ✸ P T • If we set ω p ✸ q “ 1 we will obtain the uniform measure . • Let j be the coordinate of ✸ . Set ω p ✸ q “ q ´ j , 1 ą q ą 0 . ω p T q “ const p a , b , c q ¨ q ´ volume . 4 / 35

Tiling model Tiling model Consider the probability measure on the set of tilings defined by ω p T q ź P p T q “ Z p a , b , c q , where ω p T q “ ω p ✸ q . ✸ P T • If we set ω p ✸ q “ 1 we will obtain the uniform measure . • Let j be the coordinate of ✸ . Set ω p ✸ q “ q ´ j , 1 ą q ą 0 . ω p T q “ const p a , b , c q ¨ q ´ volume . • Let j be the coordinate of ✸ . Set ω p ✸ q “ κ q j ´ κ ´ 1 q ´ j , 1 ą q ą 0 . 4 / 35

Tiling model Limit shape 5 / 35

Tiling model Waterfall Figure: A simulation for a “ 80, b “ 80, c “ 80. On the left picture the parameters are κ 2 “ ´ 1 , q “ 0 . 8, and on the right picture the parameters are κ 2 “ ´ 1 , q “ 0 . 98 . 6 / 35

Tiling model Affine transformation 7 / 35

Tiling model Tiling model It establishes a bijection between tilings and non-intersecting paths: Let C p t q “ p x 1 , x 2 , . . . , x N q be the positions of nodes. 8 / 35

Tiling model q-Racah ensemble Theorem (Borodin, Gorin, Rains ’2009) N ź ź p σ p x i q´ σ p x j qq 2 Prob t C p t q “ p x 1 , . . . , x N qu “ C ¨ w t p x i q , 0 ď i ă j ď M i “ 1 where σ p x i q “ q ´ x i ` u p κ, N , S , T q q x i and w t p x q is the weight function of the q-Racah polynomial ensemble up to a factor not depending on x. 9 / 35

Tiling model Limit shape for domino and lozenze tilings • Law of Large Numbers for the height function [Cohn–Larsen–Propp ’98], [Cohn–Kenyon–Propp ’01], [Kenyon–Okounkov ’07] 10 / 35

Tiling model Limit shape for domino and lozenze tilings • Law of Large Numbers for the height function [Cohn–Larsen–Propp ’98], [Cohn–Kenyon–Propp ’01], [Kenyon–Okounkov ’07] • Central Limit Theorem : convergence of the global fluctuations of the height function [Kenyon ’01], [Borodin–Ferrari ’08], [Petrov ’13], [Duse–Metcalfe ’14], [Bufetov–Gorin ’17], [Bufetov-K. ’17] 10 / 35

Tiling model Regularity assumptions • Let q P p 0 , 1 q , M ě 1 and u P r 0 , 1 q . 11 / 35

Tiling model Regularity assumptions • Let q P p 0 , 1 q , M ě 1 and u P r 0 , 1 q . • Let q N P p 0 , 1 q , M N P N and u N P r 0 , 1 q be sequences of parameters such that 11 / 35

Tiling model Regularity assumptions • Let q P p 0 , 1 q , M ě 1 and u P r 0 , 1 q . • Let q N P p 0 , 1 q , M N P N and u N P r 0 , 1 q be sequences of parameters such that • M N ě N ´ 1 ˇ , | M N ´ N ¨ M | , N | u N ´ u | N 2 ˇ ˇ q N ´ q 1 { N ˇ • max ` ˘ ď A 1 , for some A 1 ą 0. 11 / 35

Tiling model Regularity assumptions • Let q P p 0 , 1 q , M ě 1 and u P r 0 , 1 q . • Let q N P p 0 , 1 q , M N P N and u N P r 0 , 1 q be sequences of parameters such that • M N ě N ´ 1 ˇ , | M N ´ N ¨ M | , N | u N ´ u | N 2 ˇ ˇ q N ´ q 1 { N ˇ • max ` ˘ ď A 1 , for some A 1 ą 0. • Denote N µ N “ 1 ÿ δ p ℓ i q , N i “ 1 where p ℓ 1 , . . . , ℓ N q is P N ´ distributed with parameters q N , u N , M N . 11 / 35

Tiling model Assumptions on the weight Let w p s ; N q “ exp p´ NV N p s qq , 12 / 35

Tiling model Assumptions on the weight Let w p s ; N q “ exp p´ NV N p s qq , • V N is continuous on the intervals r 1 ` u N , q ´ M N ` u N q M N N s ; N 12 / 35

Tiling model Assumptions on the weight Let w p s ; N q “ exp p´ NV N p s qq , • V N is continuous on the intervals r 1 ` u N , q ´ M N ` u N q M N N s ; N • For some positive constants A 2 , A 3 ą 0 | V N p s q ´ V p s q| ď A 2 ¨ N ´ 1 log p N q , where V is continuous and | V p s q| ď A 3 . 12 / 35

Tiling model Assumptions on the weight Let w p s ; N q “ exp p´ NV N p s qq , • V N is continuous on the intervals r 1 ` u N , q ´ M N ` u N q M N N s ; N • For some positive constants A 2 , A 3 ą 0 | V N p s q ´ V p s q| ď A 2 ¨ N ´ 1 log p N q , where V is continuous and | V p s q| ď A 3 . • V p s q is differentiable and for some A 4 ą 0 1 ` | log | s ´ 1 ´ u || ` | log | s ´ q ´ M ´ uq M || ˇ V 1 p s q ˇ ď A 4 ¨ ˇ ˇ “ ‰ , 1 ` u , q ´ M ` uq M ‰ “ for s P . 12 / 35

Tiling model Law of Large Numbers Theorem (Dimitrov-K. ’18) There is a deterministic, compactly supported and absolutely continuous probability measure µ p x q dx such that µ N concentrate p in probability q near µ . More precisely, for any Lipschitz function f p x q defined on a real neighborhood of the interval r 1 ` u , q ´ M ` uq M s and each ε ą 0 the random variables ˇ ˇ ż ż N 1 { 2 ´ ε ˇ ˇ f p x q µ N p dx q ´ f p x q µ p x q dx ˇ ˇ ˇ R R ˇ converge to 0 in probability and in the sense of moments. 13 / 35

Tiling model Theorem (Dimitrov-K. ’18) Take m ě 1 polynomials f 1 , . . . , f m P R r x s and define „ż  ż L f i “ N f j p x q µ N p dx q´ N E P N f j p x q µ N p dx q for i “ 1 , . . . , m . R R Assume that the limit measure has one band, then under technical assumptions the random variables L f i converge jointly in the sense of moments to an m-dimensional centered Gaussian vector X “ p X 1 , . . . , X m q with covariance 1 ¿ ¿ Cov p X i , X j q “ f i p s q f j p t q C p s , t q dsdt , where p 2 π i q 2 Γ Γ ˜ ¸ 1 p s ´ α 1 qp t ´ β 1 q ` p t ´ α 1 qp s ´ β 1 q C p s , t q “ 1 ´ , 2 p s ´ t q 2 a a 2 p s ´ α 1 qp s ´ β 1 q p t ´ α 1 qp t ´ β 1 q where Γ is a positively oriented contour that encloses the interval r 1 ` u , q ´ M ` uq M s . 14 / 35

Tiling model General β Let q P p 0 , 1 q , u P r 0 , 1 q , M ě N ´ 1 P Z , θ ą 0 . • Recall “ 1 ´ q x Γ q p x q “ p 1 ´ q q 1 ´ x p q ; q q 8 and satisfies Γ q p x ` 1 q 1 ´ q , p q x ; q q 8 Γ q p x q where p y ; q q k “ p 1 ´ y q · · · p 1 ´ yq k ´ 1 q . 15 / 35

Tiling model General β Let q P p 0 , 1 q , u P r 0 , 1 q , M ě N ´ 1 P Z , θ ą 0 . • Recall “ 1 ´ q x Γ q p x q “ p 1 ´ q q 1 ´ x p q ; q q 8 and satisfies Γ q p x ` 1 q 1 ´ q , p q x ; q q 8 Γ q p x q where p y ; q q k “ p 1 ´ y q · · · p 1 ´ yq k ´ 1 q . • Let ℓ i “ q ´ λ i ` uq λ i , where λ i “ x i ` p i ´ 1 q θ, and 0 ď x 1 ď x 2 ď · · · ď x N ď M ´ N ` 1 . 15 / 35

Tiling model General β Let q P p 0 , 1 q , u P r 0 , 1 q , M ě N ´ 1 P Z , θ ą 0 . • Recall “ 1 ´ q x Γ q p x q “ p 1 ´ q q 1 ´ x p q ; q q 8 and satisfies Γ q p x ` 1 q 1 ´ q , p q x ; q q 8 Γ q p x q where p y ; q q k “ p 1 ´ y q · · · p 1 ´ yq k ´ 1 q . • Let ℓ i “ q ´ λ i ` uq λ i , where λ i “ x i ` p i ´ 1 q θ, and 0 ď x 1 ď x 2 ď · · · ď x N ď M ´ N ` 1 . • Denote X θ N : “ tp ℓ 1 , . . . , ℓ N qu . 15 / 35

Tiling model General β We consider probability measures on X θ N P N θ p ℓ 1 , . . . , ℓ N q “ 1 q 2 θλ j Γ q p λ j ´ λ i ` 1 q Γ q p λ j ´ λ i ` θ q ź Γ q p λ j ´ λ i q Γ q p λ j ´ λ i ` 1 ´ θ q × Z N 1 ď i ă j ď N N Γ q p λ j ` λ i ` v ` 1 q Γ q p λ j ` λ i ` v ` θ q ź ź × Γ q p λ j ` λ i ` v q Γ q p λ j ` λ i ` v ` 1 ´ θ q ¨ w p ℓ i q , 1 ď i ă j ď N i “ 1 v is such that q u “ v . 16 / 35

Tiling model General β analogue Observe as q Ñ 1 ´ and q x Ñ y P r 0 , 1 q p q x ; q q Γ q p x ` α q “ p 1 ´ q q ´ α „ p 1 ´ q q ´ α p 1 ´ y q α . p q x ` α ; q q 8 Γ q p x q 17 / 35