The traffic distribution of the squared unimodular random matrix and a formula for the moments of its ESD

TL;DR

This paper derives a polynomial formula for the moments of the empirical spectral distribution of squared unimodular random matrices using traffic-free probability, disproving a previous conjecture.

Contribution

It introduces a novel approach to compute spectral moments via traffic-free probability and provides explicit polynomial coefficients in terms of graph quotients.

Findings

Polynomial expression for spectral moments involving graph quotients

Disproof of the previously conjectured formula

Application of traffic-free probability to random matrix spectra

Abstract

The $k$-th moment of the mean empirical spectral distribution of the squared unimodular random matrix of dimension $N$ can be expressed in the form $N^{-2k-1} Q_k(N)$, where $Q_k(x)$ is a polynomial of degree $k+1$ with integer coefficients. We use tools from traffic-free probability to express the coefficients of this polynomial in terms of the number of quotients, with a certain property, of some colored directed graphs. The obtained result disproves the formula conjectured in A. Lakshminarayan, Z. Puchala, K. Zyczkowski (2014).