Statistical mechanics via Answers: GUE asymptotics of symmetric - PowerPoint PPT Presentation

Asymptotics of symmetric functions Greta Panova Lozenge tilings The objects Probabilistic questions Statistical mechanics via Answers: GUE asymptotics of symmetric functions Probability via Schur functions Schur functions asymptotics GUE proof Greta Panova (University of Pennsylvania) Tilings with free boundaries ASMs and GUE based on: Dense loop model 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 . April 2015 1

Asymptotics of Overview symmetric functions Alternating Sign Matrices Greta Panova (ASM)/ 6-Vertex model: Characters of U ( ∞ ), boundary Lozenge tilings   The objects of the Gelfand-Tsetlin graph 0 0 1 0 Probabilistic questions   1 1 1 2 2 . . . − 1 0 1 1 Answers: GUE     1 − 1 1 0 2 2 3 . . . Probability via Schur functions 0 1 0 0 . . . Schur functions asymptotics GUE proof Normalized Schur functions: Tilings with free boundaries S λ ( x 1 , . . . , x k ; N ) = s λ ( x 1 , . . . , x k , 1 N − k ) ASMs and GUE s λ (1 N ) Dense loop model Lozenge tilings: Dense loop model: ζ 1 ζ 2 y x L 2

Asymptotics of Lozenge tilings symmetric functions Greta Panova Tilings of a domain Ω (on a Lozenge tilings The objects triangular lattice) with elementary Probabilistic questions rhombi of 3 types (“lozenges”). Answers: GUE Probability via Schur functions Schur functions asymptotics GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 3

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 Probability via Schur 3 2 2 1 functions Schur functions 2 1 1 1 asymptotics 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 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

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 Answers: GUE 5 4 3 2 2 1 Probability via Schur 5 3 2 2 1 functions Schur functions 2 1 1 1 4 asymptotics 4 4 1 1 GUE proof 4 Tilings with free 3 boundaries 3 3 3 3 ASMs and GUE Dense loop model 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 4

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of The many faces of lozenge tilings symmetric functions Greta Panova Lozenge tilings The objects 5 4 4 4 3 2 5 4 4 4 3 2 Probabilistic questions 5 3 3 2 2 1 5 3 3 2 2 1 Answers: GUE 4 3 2 2 1 4 3 2 2 1 Probability via Schur 3 2 2 1 3 2 2 1 functions Schur functions 2 1 1 1 2 1 1 1 asymptotics 1 1 1 1 GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 5

Asymptotics of Limit behavior symmetric functions Greta Panova Question: Fix Ω in the plane and let grid size → 0, what are the properties of uniformly random tilings of Ω? Lozenge tilings The objects Probabilistic questions [Cohn–Larsen–Propp, 1998] Hexagonal domain: Tiling is asymptotically frozen Answers: GUE outside inscribed ellipse. Probability via Schur functions [Kenyon–Okounkov, 2005] Polygonal domain: Tiling is asymptotically frozen Schur functions outside inscribed algebraic curve. asymptotics GUE proof Tilings with free boundaries ASMs and GUE Dense loop model 2.jpg [Cohn–Kenyon–Propp, 2001; Kenyon-Okounkov-Sheffield, 2006] There exists a “limit shape” for the surface of the height function (plane partition). 6

Asymptotics of Behavior near the boundary, interlacing particles symmetric functions Greta Panova x Lozenge tilings The objects Probabilistic questions x 3 x 2 1 Answers: GUE 1 Probability via Schur N functions x 1 Schur functions 1 asymptotics x 3 GUE proof 2 Tilings with free x 2 boundaries 2 x 3 3 ASMs and GUE Dense loop model Horizontal lozenges near a flat Question: What is the joint distribution boundary: i =1 as N → ∞ (scale = 1 of { x i j } k N )? interlacing particle configuration ↔ Gelfand-Tsetlin patterns. x 1 ≤ 1 ≤ x 2 x 2 ≤ 2 ≤ ≤ 1 ≤ x 3 x 3 x 3 3 2 1 1 with an explanation what the answer should be. 2 Subsequent results: [Gorin-P,2012], [Novak, 2014],[Mkrtchyan, 2013, periodic weights, unbounded 7 region]

Asymptotics of Behavior near the boundary, interlacing particles symmetric functions Greta Panova x Lozenge tilings The objects Probabilistic questions x 3 x 2 1 Answers: GUE 1 Probability via Schur N functions x 1 Schur functions 1 asymptotics x 3 GUE proof 2 Tilings with free x 2 boundaries 2 x 3 3 ASMs and GUE Dense loop model Horizontal lozenges near a flat Question: What is the joint distribution boundary: i =1 as N → ∞ (scale = 1 of { x i j } k N )? interlacing particle configuration Conjecture ↔ Gelfand-Tsetlin patterns. [Okounkov–Reshetikhin, 2006 1 ]: x 1 The joint distribution converges to a ≤ 1 ≤ GUE -corners (aka GUE -minors) process: x 2 x 2 ≤ 2 ≤ ≤ 1 ≤ eigenvalues of GUE matrices. x 3 x 3 x 3 3 2 1 Proven for the hexagon by Johansson- Nordenstam (2006). 2 1 with an explanation what the answer should be. 2 Subsequent results: [Gorin-P,2012], [Novak, 2014],[Mkrtchyan, 2013, periodic weights, unbounded 7 region]