Freely Independent Coin Tosses, Standard Young Tableaux, and the Kesten--McKay Law

TL;DR

This paper connects random walks on regular trees, the Kesten-McKay law, and standard Young tableaux, using free probability to model these phenomena in random matrix theory.

Contribution

It establishes a novel link between the moments of the Kesten-McKay law and counting standard Young tableaux with at most two rows, and explores properties beyond integer degrees.

Findings

Moments of the Kesten-McKay law correspond to counting standard Young tableaux.

Properties of the walk extend to non-integer degrees.

Free probability provides a framework for modeling in random matrix theory.

Abstract

In this article, we shall start with a closed walk on a regular tree of degree $d$. These walks are described by the Kesten-McKay law which arises as the asymptotic distribution of a random $d$-regular graph on $n$ vertices. We will show that the moments of the Kesten-McKay law are given by counting standard Young tableaux with at most 2 rows, and how some properties of the walk make sense even when $d$ is not an integer. We will use free probability to instruct us how to build an explicit model in random matrix theory.