REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: Various asymptotic aspects of the Hook Length Formula for standard Young tableaux have been studied recently in combinatorics and probability. In this talk, we study the limiting distributions that come from random variables associated to Stanley’s q-hook-content formula for semistandard tableaux and q-hook length formulas of Björner–Wachs related to linear extensions of labeled forests. We show that, while these limiting distributions are “generically” asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the moduli space of distributions of these statistics in several regimes. The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded 2-norm. The closure of the moduli space of these distributions in the Lévy metric gives rise to the moduli space of DUSTPAN distributions. As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box. This talk is based on joint work with Joshua Swanson at UCSD (https://arxiv.org/abs/2010.12701).