Symmetric groups and random matrices

TL;DR

This paper introduces a combinatorial framework linking symmetric group conjugacy class products to random matrix theory, enabling exact formulas and precise asymptotic analysis of characters and measures on Young diagrams.

Contribution

It develops a new combinatorial approach that simplifies manipulation of conjugacy class products and connects them to random matrix techniques, providing exact and asymptotic results.

Findings

Exact formulas for products of conjugacy class indicators

Asymptotic analysis of characters of large symmetric groups

Asymptotics of the Plancherel measure on Young diagrams

Abstract

The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily and we show that it very closely related to the combinatorial approach to random matrices. Our formulas are exact (in a sense that they hold not only asymptotically for large q). This result has many interesting applications, for example it allows to find precise asymptotics of characters of large symmetric groups and asymptotics of the Plancherel measure on Young diagrams.