Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory

Abstract

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their $q$-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit. We also investigate similar behavior for alternating sign matrices (equivalently, six-vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in $O(n=1)$ dense loop model.

Published at http://dx.doi.org/10.1214/14-AOP955 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• From Alternating Sign Matrices to the Gaussian Unitary Ensemble
• Exact spin quantum Hall current between boundaries of a lattice strip
• Separation of variables for symplectic characters
• Lozenge tilings with free boundaries

Cited by in corpus (13)

• Stochastic six-vertex model
• Fluctuations of particle systems determined by Schur generating functions
• Fourier transform on high-dimensional unitary groups with applications to random tilings
• Bulk universality for random lozenge tilings near straight boundaries and for tensor products
• Quantized Vershik-Kerov Theory and Quantized Central Measures on Branching Graphs
• Matrix addition and the Dunkl transform at high temperature
• A statistical mechanical approach to restricted integer partition functions
• Product matrix processes as limits of random plane partitions
• Arctic curves of the four-vertex model
• Lozenge tilings with free boundaries
• Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions
• On the logarithmic bipartite fidelity of the open XXZ spin chain at $\Delta =-1/2$
• Characters of classical groups, Schur-type functions, and discrete splines