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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38 att a Yhfh Pf-.,. Cc.r f,'lrqouJ c"to Fourrar T\r'*l Frr.',p/a ' Foe.reF r. hFh

t tts tr r

Tiling Shuffling Phenomenon Tri Lai University of Nebraska Lincoln Lincoln, NE 68588 Dimers

Asymptotics of symmetric functions with applications to Setup Asymptotics of statistical

analytic limit shapes for the 5-vertex model Richard Kenyon (Brown University) Based on joint

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,

Exactly solvable models of tilings and LittlewoodRichardson coefficients P. Zinn-Justin

LOZENGE TILINGS WITH GAPS IN A 90 WEDGE DOMAIN WITH MIXED BOUNDARY CONDITIONS Mihai Ciucu

Statistical mechanics via Answers: GUE asymptotics of symmetric functions Probability via Schur

Tilings and discrete geometry V. Berth e, T. Fernique LIRMM-CNRS-Montpellier-France

Effective Hamiltonians of anyon anyon lattice lattice Effective Hamiltonians of models models

A matrix model for counting plane partitions and tilings Bertrand Eynard, IPHT CEA-SACLAY

Domino tilings, lattice paths and plane overpartitions Sylvie Corteel LIAFA, CNRS et Universit

Random tilings and representations of classical Lie groups Alexey Bufetov University of Bonn 21

Boolean Lattice and Symmetric Chain Decompositions Yizhe Zhu Shanghai Jiao Tong University

Random matrices and Aztec diamonds Kurt Johansson Florence, May, 2015. Domino Tilings of the

Fractal Tilings Katie Moe and Andrea Brown December 13, 2006 Introduction Examples of Fractal

Lattice and Non-Lattice Markov Additive Models Jevgenijs Ivanovs, Guy Latouche and Peter Taylor

Fixed point aperiodic tilings Bruno Durand, Andrei Romashchenko, Alexander Shen LIF Marseille,

Lets count: Domino tilings Christopher R. H. Hanusa Queens College, CUNY 2 n 3 n n n

Area under lattice paths associated with certain urn models. Alois Panholzer, Markus Kuba

Software to Produce Fractal Tilings of the Plane University of Wollongong Dr. Samuel Webster and

Clients Matter, Services Don't Recovering the Future of Services on the Web Mike Amundsen, API

Probing Baryogenesis with Displaced Vertices at the LHC Yanou Cui Perimeter Institute

Industry Day Susan Johnson January 22, 2015 Research and Technologies for Aerospace Propulsion

38 att a Yhfh Pf-.,. Cc.r f,'lrqouJ c&quot;to Fourrar T\r'*l Frr.',p/a ' Foe.reF r. hFh

t tts tr r

Tiling Shuffling Phenomenon Tri Lai University of Nebraska Lincoln Lincoln, NE 68588 Dimers

Asymptotics of symmetric functions with applications to Setup Asymptotics of statistical

analytic limit shapes for the 5-vertex model Richard Kenyon (Brown University) Based on joint

38 att a Yhfh Pf-.,. Cc.r f,'lrqouJ c"to Fourrar T\r'*l Frr.',p/a ' Foe.reF r. hFh