对称多项式的渐近性及其在统计力学和表示论中的应用
Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
摘要：
我们开发了一种新方法来研究变量数趋于无穷大时表示论起源的对称多项式的渐进性。给出了方法的几个应用：证明了关于无限维酉群的特征及其$q$ -变形的若干定理。本文研究了大多边形区域上均匀随机菱形镶嵌的性质，并在极限下找到了GUE-特征值分布。我们还研究了交替符号矩阵（相当于，具有域壁边界条件的六顶点模型）的类似行为。最后，我们计算了$O(n=1)$稠密环模型中某些观测量的渐近展开式。
英文摘要：
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 $O(n=1)$ dense loop model.
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