Asymptotics of symmetric functions with applications to Setup - PowerPoint PPT Presentation

Asymptotics of symmetric functions with applications to statistical mechanics and representation theory Greta Panova (UCLA) Normalized Schur functions S λ Asymptotics of symmetric functions with applications to Setup Asymptotics of statistical mechanics and representation theory S λ ( N )( x 1 , . . . , xk ) GUE in random lozenge tilings Lozenge tilings N → ∞ , behavior Greta Panova (UCLA) near boundary GUE GUE in tilings, results based on same-name paper arXiv:1301.0634 joined with Vadim Gorin ASM GUE in ASMs FPSAC 2013, Paris 1

Asymptotics of Overview symmetric functions with applications to statistical mechanics Alternating Sign Matrices and representation theory (ASM)/ 6Vertex model: Characters of U ( ∞ ), boundary Greta Panova (UCLA) of the Gelfand-Tsetlin graph   0 0 1 0 1 1 1 2 2 . . . Normalized Schur   0 1 − 1 1 functions S λ   2 2 3 . . .   1 − 1 1 0 Setup . . . Asymptotics of 0 1 0 0 S λ ( N )( x 1 , . . . , xk ) GUE in random lozenge tilings Normalized Schur functions: Lozenge tilings N → ∞ , behavior S λ ( x 1 , . . . , x k ; N ) = s λ ( x 1 , . . . , x k , 1 N − k ) near boundary GUE s λ (1 N ) GUE in tilings, results ASM GUE in ASMs Lozenge tilings: Dense loop model: ζ 1 y ζ 2 x L 2

Asymptotics of Definitions and setup symmetric functions with applications to statistical mechanics In our context: Symmetric functions, Lie groups characters. and representation theory Greta Panova (UCLA) Normalized Schur functions S λ Setup Asymptotics of S λ ( N )( x 1 , . . . , xk ) GUE in random lozenge tilings Lozenge tilings (mainly) Schur functions: s λ ( x 1 , . . . , x N ) – characters of V λ . N → ∞ , behavior near boundary GUE GUE in tilings, results ASM GUE in ASMs 3

Asymptotics of Definitions and setup symmetric functions with applications to statistical mechanics In our context: Symmetric functions, Lie groups characters. and representation theory Irreducible (rational) representations V λ of GL ( N ) (or U ( N )) are indexed by Greta Panova (UCLA) dominant weights (signatures/Young diagrams/integer partitions) λ : Normalized Schur λ 1 ≥ λ 2 ≥ · · · ≥ λ N , functions S λ Setup Asymptotics of where λ i ∈ Z , e.g. λ = (4 , 3 , 1) , S λ ( N )( x 1 , . . . , xk ) GUE in random lozenge tilings Lozenge tilings (mainly) Schur functions: s λ ( x 1 , . . . , x N ) – characters of V λ . N → ∞ , behavior near boundary GUE GUE in tilings, results ASM GUE in ASMs 3

Asymptotics of Definitions and setup symmetric functions with applications to statistical mechanics In our context: Symmetric functions, Lie groups characters. and representation theory Irreducible (rational) representations V λ of GL ( N ) (or U ( N )) are indexed by Greta Panova (UCLA) dominant weights (signatures/Young diagrams/integer partitions) λ : Normalized Schur λ 1 ≥ λ 2 ≥ · · · ≥ λ N , functions S λ Setup Asymptotics of where λ i ∈ Z , e.g. λ = (4 , 3 , 1) , S λ ( N )( x 1 , . . . , xk ) GUE in random lozenge tilings Lozenge tilings (mainly) Schur functions: s λ ( x 1 , . . . , x N ) – characters of V λ . N → ∞ , behavior near boundary Weyl’s determinantal formula: GUE GUE in tilings, results � � N λ j + N − j ASM det x i GUE in ASMs ij =1 s λ ( x 1 , . . . , x N ) = � i < j ( x i − x j ) Semi-Standard Young tableaux ( ⇔ Gelfand-Tsetlin patterns) of shape λ : ( x 1 , x 2 , x 3 ) = x 2 1 x 2 + x 2 1 x 2 + x 2 2 x 2 + x 2 + x 1 x 2 + x 1 x 2 x 2 s (2 , 2) ( x 1 , x 2 , x 3 ) = s 1 x 2 x 3 2 x 3 . 2 3 3 3 1 1 1 1 2 2 1 1 1 2 1 2 2 2 3 3 3 3 2 3 2 3 3 3 3

Asymptotics of Definitions and setup symmetric functions with applications to statistical mechanics Object of study and main tool in the applications: and representation theory Normalized Schur functions: Greta Panova (UCLA) N − k � �� � Normalized Schur s λ ( N ) ( x 1 , . . . , x k , 1 , . . . , 1) functions S λ S λ ( N ) ( x 1 , . . . , x k ) = Setup s λ ( N ) (1 , . . . , 1 ) Asymptotics of � �� � S λ ( N )( x 1 , . . . , xk ) N GUE in random lozenge tilings Fix k , let N → ∞ and let Lozenge tilings � i � N → ∞ , behavior λ ( N ) i near boundary → f GUE N N GUE in tilings, results ASM Limit shape of λ ( N ) is f ( t ): GUE in ASMs 0 1 λ ( N ) f ( t ) N f ( t ) 1 4

Asymptotics of Integral formula, k = 1 asymptotics symmetric functions with applications to statistical mechanics and representation Theorem (G–P) theory For any signature λ ∈ GT N and any x ∈ C other than 0 or 1 we have Greta Panova (UCLA) � x z ( N − 1)! 1 Normalized Schur S λ ( x ; N , 1) = dz , � N functions S λ ( x − 1) N − 1 2 π i i =1 ( z − ( λ i + N − i )) C Setup Asymptotics of S λ ( N )( x 1 , . . . , xk ) where the contour C includes all the poles of the integrand. GUE in random (Similar statements hold for a larger class of functions, e.g symplectic lozenge tilings characters, Jacobi...also q –analogues; formula appears also in Lozenge tilings N → ∞ , behavior [Colomo,Pronko,Zinn-Justin]) near boundary � i � GUE Let λ ( N ) i → f under certain convergence conditions... GUE in tilings, results N N ASM using the method of steepest descent we obtain various asymptotic formula: GUE in ASMs Theorem (G–P) Under [certain strong convergence conditions of] λ ( N ) towards the limit shape N f , as N → ∞ : � � S λ ( N ) ( e y ; N , 1) = G ( w 0 , f ) exp( N ( yw 0 − F ( w 0 ; f ))) 1 + o (1) , e N ( e y − 1) N − 1 � 1 ∂ where F ( w ; f ) = 0 ln( w − f ( t ) − 1 + t ) dt, w 0 is the root of ∂ w F ( w ; f ) = y (inverse Hilbert transform) and G is a certain explicit function. 5

Asymptotics of Integral formula, k = 1 asymptotics symmetric functions with applications to statistical mechanics and representation Theorem (G–P) theory For any signature λ ∈ GT N and any x ∈ C other than 0 or 1 we have Greta Panova (UCLA) � x z ( N − 1)! 1 Normalized Schur S λ ( x ; N , 1) = dz , � N functions S λ ( x − 1) N − 1 2 π i i =1 ( z − ( λ i + N − i )) C Setup Asymptotics of S λ ( N )( x 1 , . . . , xk ) where the contour C includes all the poles of the integrand. GUE in random (Similar statements hold for a larger class of functions, e.g symplectic lozenge tilings characters, Jacobi...also q –analogues; formula appears also in Lozenge tilings N → ∞ , behavior [Colomo,Pronko,Zinn-Justin]) near boundary � i � GUE Let λ ( N ) i → f under certain convergence conditions... GUE in tilings, results N N ASM using the method of steepest descent we obtain various asymptotic formula: GUE in ASMs Theorem (G–P) Under [some other convergence conditions of] λ ( N ) towards the limit shape f , N as N → ∞ � √ � √ NE ( f ) h + 1 2 S ( f ) h 2 + o (1) S λ ( N ) ( e h / N ; N , 1) = exp , � 1 � 1 � 1 f ( t ) 2 dt − E ( f ) 2 + f ( t )(1 − 2 t ) dt . where E ( f ) = f ( t ) dt , S ( f ) = 0 0 0 5

Asymptotics of From k = 1 asymptotics to general k , multiplicativity symmetric functions with applications to statistical mechanics and representation Theorem (G–P) theory For any signature λ ∈ GT N and any k ≤ N we have Greta Panova (UCLA) Normalized Schur N − k � �� � functions S λ k � S λ ( x 1 , . . . , x k ; N ) = s λ ( x 1 , . . . , x k , 1 , . . . , 1) ( N − i )! Setup = ( N − 1)!( x i − 1) N − k × Asymptotics of s λ (1 , . . . , 1 ) S λ ( N )( x 1 , . . . , xk ) i =1 � �� � GUE in random N lozenge tilings � � k Lozenge tilings D j − 1 det N → ∞ , behavior k � i , 1 near boundary i , j =1 S λ ( x j ; N , 1)( x j − 1) N − 1 . GUE ∆( x 1 , . . . , x k ) GUE in tilings, results j =1 ASM ∂ GUE in ASMs where D i , 1 = x i ∂ x i and ∆ – Vandermonde determinant. Similar theorems for symplectic characters, Jacobi; also q -analogues (replacing derivatives by q -shifts). Note: appears in [de Gier, Nienhuis, Ponsaing] for symplectic characters. 6

Asymptotics of From k = 1 asymptotics to general k , multiplicativity symmetric functions with applications to statistical mechanics Theorem (G–P) and representation theory For any signature λ ∈ GT N and any k ≤ N we have Greta Panova (UCLA) N − k � �� � k � Normalized Schur S λ ( x 1 , . . . , x k ; N ) = s λ ( x 1 , . . . , x k , 1 , . . . , 1) ( N − i )! = ( N − 1)!( x i − 1) N − k × functions S λ s λ (1 , . . . , 1 ) i =1 Setup � �� � Asymptotics of N S λ ( N )( x 1 , . . . , xk ) � � k D j − 1 det k � GUE in random i , 1 i , j =1 S λ ( x j ; N , 1)( x j − 1) N − 1 . lozenge tilings ∆( x 1 , . . . , x k ) Lozenge tilings j =1 N → ∞ , behavior near boundary ∂ where D i , 1 = x i ∂ x i and ∆ – Vandermonde determinant. GUE GUE in tilings, results Corollary (G–P) ASM Suppose that the sequence λ ( N ) is such that GUE in ASMs � � ln S λ ( N ) ( x ; N , 1) lim = Ψ( x ) N →∞ N uniformly on compact subsets of a region M ⊂ C (e.g. Theorem 2). Then � � ln S λ ( N ) ( x 1 , . . . , x k ; N , 1) lim = Ψ( x 1 ) + · · · + Ψ( x k ) N N →∞ for any k uniformly on compact subsets of M k . I.e., informally, under various regimes of convergence for λ ( N ) we have S λ ( N ) ( x 1 , . . . , x k ) ≃ S λ ( N ) ( x 1 ) · · · S λ ( N ) ( x k ) 6