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

Abstract

Abstract. We develop a new method for studying the asymptotics of symmetric polynomials of representation–theoretic origin as the num-ber 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 do-mains 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)