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We develop a new method for studying the asymptotics of symmetric polynomialsof representation-theoretic origin as the number of variables tends toinfinity. Several applications of our method are presented: We prove a numberof theorems concerning characters of infinite-dimensional unitary group andtheir $q$-deformations. We study the behavior of uniformly random lozengetilings of large polygonal domains and find the GUE-eigenvalues distribution inthe 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 $O(n=1)$ denseloop model.

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