Lozenge tilings and other lattice models via symmetric functions - PowerPoint PPT Presentation

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model Lozenge tilings and other lattice models – via symmetric functions Greta Panova (University of Pennsylvania) based on: V.Gorin, G.Panova, Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory, Annals of Probability arXiv:1301.0634 G. Panova, Lozenge tilings with free boundaries, arXiv:1408.0417 . Firenze, Maggio 2015 1

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model Overview Alternating Sign Matrices (ASM)/ 6-Vertex model: Characters of U ( 1 ), boundary 0 1 of the Gelfand-Tsetlin graph 0 0 1 0 B C 1 1 1 2 2 . . . 0 1 � 1 1 B C @ A 1 � 1 1 0 2 2 3 . . . 0 1 0 0 . . . Normalized Schur functions: S � ( x 1 , . . . , x k ; N ) = s � ( x 1 , . . . , x k , 1 N � k ) s � (1 N ) Lozenge tilings: Dense loop model: ζ 1 ζ 2 y x L 2

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model Lozenge tilings Tilings of a domain Ω (on a triangular lattice) with elementary rhombi of 3 types (“lozenges”). 3

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model The many faces of lozenge tilings 5 4 4 4 3 2 5 3 3 2 2 1 4 3 2 2 1 3 2 2 1 2 1 1 1 1 1 5 5 4 4 3 4 3 3 3 4 2 2 2 2 3 1 1 2 2 2 2 1 1 1 1 1 1 4

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model Combinatorics: how many? 5 4 4 4 3 2 5 3 3 2 2 1 4 3 2 2 1 3 2 2 1 2 1 1 1 1 1 [MacMahon]: Boxed plane partitions (tilings of a ⇥ b ⇥ c ⇥ a ⇥ b ⇥ c hexagon) a b c Y Y Y i + j + k � 1 = i + j + k � 2 i =1 j =1 k =1 General: Lindstr¨ om-Gessel-Viennot determinants; hook-content formula. 6

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model Probability: limit behavior Question: Fix Ω in the plane and let grid size ! 0, what are the properties of uniformly random tilings of Ω ? 2.jpg Frozen regions (polygonal domains), “limit shapes” of the surface of the height function (plane partitions). ([Cohn–Larsen–Propp, 1998], [Kenyon–Okounkov, 2005], [Cohn–Kenyon–Propp, 2001; Kenyon-Okounkov-She ffi eld, 2006] ) 7

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model Behavior near the boundary, interlacing particles x x 3 1 x 2 1 N x 1 1 x 3 2 x 2 2 x 3 3 Question: Joint distribution of { x i j } k i =1 as N ! 1 Horizontal lozenges near a flat (rescaled)? boundary: x 1 1   x 2 x 2  2   1  x 3 x 3 x 3 3 2 1 8

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model Behavior near the boundary, interlacing particles x x 3 1 x 2 1 N x 1 1 x 3 2 x 2 2 x 3 3 Question: Joint distribution of { x i j } k i =1 as N ! 1 Horizontal lozenges near a flat (rescaled)? boundary: Conjecture [Okounkov–Reshetikhin, 2006]: x 1 1   The joint distribution converges to a GUE -corners x 2 x 2 (aka GUE -minors) process: eigenvalues of GUE ma-  2   1  trices. x 3 x 3 x 3 3 2 1 Proofs: hexagonal domain [Johansson-Nordenstam, 2006], more general domains [Gorin-P,2012], [Novak, 2014], un- bounded [Mkrtchyan, 2013] 8

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model The Gaussian Unitary Ensemble (GUE) GUE: matrices [ X ij ] i , j : X = X T Re X ij , Im X ij – i.i.d. ⇠ N (0 , 1 / 2), i 6 = j X ii – i.i.d. ⇠ N (0 , 1) 0 1 a 11 a 12 a 13 a 14 ( x k 1 � x k 2 � · · · � x k k ) – eigenvalues of [ X i , j ] k i , j =1 B C a 21 a 22 a 23 a 24 B C @ A a 31 a 32 a 33 a 34 x j i � 1  x j � 1 i � 1  x j Interlacing condition: a 41 a 42 a 43 a 44 i x 4 x 4 x 4 x 4 1 2 3 4 x 3 x 3 x 3 1 2 3  x 2 x 2  1 2 x 1 1 The joint distribution of { x j i } 1  i  j  k is the GUE–corners (also, GUE-minors) process , =: GUE k . 9

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model GUE in tilings: our setup Domain Ω � ( N ) : positions of the N horizontal lozenges on right boundary are: � ( N ) 1 + N � 1 > � ( N ) 2 + N � 2 > · · · > � ( N ) N 8 + 10 8 + 9 +4 8 + 8 1 8 + 7 8 + 6 2 +3 3 +2 +1 4 0 + 5 0 + 4 0 + 3 5 0 + 2 � (5) = (4 , 3 , 3 , 0 , 0) 0 + 1 ( 1 0 + 0 N Ω � ( N ) is not necessarily a fi- � = ( a , . . . , a , 0 , . . . , 0 ) nite polygon as N ! 1 , e.g. | {z } | {z } � ( N ) = ( N , N � 1 , . . . , 2 , 1)) c b $ a ⇥ b ⇥ c ... hexagon. 10

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model Plane partitions/Gelfand-Tsetlin patterns λ 1 + N − 1 x 3 λ 2 + N − 2 1 x 3 2 x 3 3 λ N Line j = 3 5 4 4 4 3 3 4 3 3 3 3 3 N 3 3 2 1 3 1 0 0 2 1 0 0 0 2 0 0 0 0 � = (4 , 3 , 3 , 0 , 0) x 3 = (4 , 3 , 0) � = (5 , 4 , 3 , 1 , 0) n o k x i Question: Joint distribution of the (rescaled) positions i =1 as N ! 1 ? j 11

Lozenge tilings Probability via Schur functions Free boundary 6-vertex/ASMs Dense loop model GUE in tilings: our results Limit profile f ( t ) of � ( N ) as N ! 1 : ✓ i ◆ � ( N ) i Theorem (Gorin-P (2012), Novak (2014)) ! f N N Let � ( N ) = ( � 1 ( N ) � . . . � � N ( N )) , N = 1 , 2 , . . . . If 9 a piecewise-di ff erentiable weakly decreasing function f ( t ) � ✓ i ◆� N X p � � � i ( N ) � ( N ) � � (limit profile of � ( N ) ) s.t. � f � = o ( N ) � N N f ( t ) N i =1 as N ! 1 and sup i , N | � i ( N ) / N | < 1 . � ( N ) = { x j Let Υ k i } k j =1 . Then 8 fixed k , as N ! 1 Ω � ( N ) domain: Υ k � ( N ) � NE ( f ) p ! GUE k (GUE-corners proc. rank k ) NS ( f ) λ 1 + N − 1 in the sense of weak convergence, where x 3 λ 2 + N − 2 1 Z 1 x 3 2 E ( f ) = f ( t ) dt , 0 Z 1 ✓ ◆ 2 f ( t ) � t + 1 dt � 1 6 � E ( f ) 2 S ( f ) = x 3 3 2 0 λ N Line j = 3 12

Lozenge tilings and other lattice models via symmetric functions - PowerPoint PPT Presentation

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