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

Vadim Gorin; Greta Panova

doi:10.1214/14-aop955

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arXiv · 1301.0634

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

Vadim Gorin·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.

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BibTeX RIS

Vadim Gorin, Greta Panova. 2015-12-21. Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. https://doi.org/10.1214/14-aop955

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