The Annals of Probability

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

Vadim Gorin and Greta Panova

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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
Received: July 2013
Revised: July 2014
First available in Project Euclid: 11 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1449843626

Digital Object Identifier
doi:10.1214/14-AOP955

Mathematical Reviews number (MathSciNet)
MR3433577

Zentralblatt MATH identifier
06541353

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 22E99: None of the above, but in this section 05E05: Symmetric functions and generalizations 60F99: None of the above, but in this section

Keywords
Symmetric polynomials Schur function lozenge tilings GUE ASM 6 vertex model dense loop model extreme characters of $U(\infty)$

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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