Z-Measures on partitions, Robinson-Schensted-Knuth correspondence, and beta=2 random matrix ensembles

TL;DR

This paper reviews the connection between combinatorial and representation theory asymptotics with beta=2 random matrix ensembles, unifying known results through generalized regular representations of the infinite symmetric group.

Contribution

It introduces a unifying framework showing that all known beta=2 ensemble results are degenerations of a single general situation involving infinite symmetric group representations.

Findings

All known results are special cases of a general degenerative framework.

The framework is based on generalized regular representations of the infinite symmetric group.

Connections between combinatorics, representation theory, and random matrix theory are systematically unified.

Abstract

We suggest an hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with ``beta=2 ensembles'' arising in the random matrix theory. We show that all such results are, essentially, degenerations of one general situation arising from so-called generalized regular representations of the infinite symmetric group.