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 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]

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

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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

Entropy production and steady states in quantum statistical mechanics Vojkan Jaksic and

From unitary dynamics to statistical mechanics in isolated quantum systems Marcos Rigol

Overview 1. Introduction to statistical mechanics: Partition functions and statistical ensembles

A Winter Short Course on Statistical Mechanics for Molecular Simulations Lecture 2: Molecular

Statistical mechanics of fitness landscapes Joachim Krug Institute for Theoretical Physics,

Einsteins approach to Statistical Mechanics A prelude to the Marvelous Year Luca Peliti In

Irreversibility and the Second Law in Thermodynamics and Statistical Mechanics Erik Curiel

to Bernard Jancovici in memoriam Analytic results in statistical mechanics Can the

On the foundations of non-equilibrium quantum statistical mechanics Vojkan Jaksic McGill

New Results for 2D Tilings: Wiener Index &amp; Statistical Mechanics of Graphs ArizMATYC &amp;

Statistical mechanics for the phase separation of interacting self propelled particles J. Barr

Statistical Mechanics of Jamming of frictional grains Dapeng ( Max ) Bi &amp; Bulbul Chakraborty

(quantum statistics) Classical statistical mechanics 1. Microcanonical ensemble = const

Replica method: a statistical mechanics approach to probability-based information processing

Spectral zeta function &amp; quantum statistical mechanics on Sierpinski carpets Joe P. Chen

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Statistical mechanics, the partition function, and fj rst- order phase transitions Magnus

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Statistical Mechanics of Jamming of frictional grains Dapeng ( Max ) Bi & Bulbul Chakraborty

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