## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 43, Number 6 (2015), 3052-3132.

Dates
Revised: July 2014
First available in Project Euclid: 11 December 2015

https://projecteuclid.org/euclid.aop/1449843626

Digital Object Identifier
doi:10.1214/14-AOP955

Mathematical Reviews number (MathSciNet)
MR3433577

Zentralblatt MATH identifier
06541353

#### Citation

Gorin, Vadim; Panova, Greta. Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. Ann. Probab. 43 (2015), no. 6, 3052--3132. doi:10.1214/14-AOP955. https://projecteuclid.org/euclid.aop/1449843626

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