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

TL;DR

This paper introduces a new method for analyzing the asymptotic behavior of symmetric polynomials of representation-theoretic origin, with applications spanning statistical mechanics, representation theory, and combinatorics.

Contribution

The paper develops a novel approach to asymptotics of symmetric polynomials and applies it to characters of infinite-dimensional groups, tiling models, and loop models.

Findings

Distribution of GUE eigenvalues in large lozenge tilings

Asymptotic behavior of characters of infinite-dimensional unitary groups

Asymptotics of observables in the dense loop model

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.