ASYMPTOTICS OF SYMMETRIC POLYNOMIALS WITH APPLICATIONS TO STATISTICAL MECHANICS AND REPRESENTATION THEORY

• 成果类型: Article
• 署名作者: Gorin, Vadim; Panova, Greta
• 署名单位: Massachusetts Institute of Technology (MIT); Kharkevich Institute for Information Transmission Problems of the RAS; Russian Academy of Sciences; University of Pennsylvania
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP955
• 发表日期: 2015
• 页码: 3052-3132
• 关键词: alternating sign matrices; gelfand-tsetlin graph; refined enumeration; variables goes; characters; BOUNDARY; number; PROOF; wall

摘要

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 Omicron (n = 1) dense loop model.