A random walk approach to linear statistics in random tournament ensembles

TL;DR

This paper studies linear statistics of random matrices with imaginary Bernoulli entries, modeling random tournaments, and shows their convergence to Gaussian distributions via a random walk approach and Ornstein-Uhlenbeck processes.

Contribution

It introduces a novel random walk method to analyze linear statistics in random tournament ensembles and establishes convergence rates to Gaussian limits.

Findings

First $k$ traces converge to Ornstein-Uhlenbeck process.

Rate of convergence to Gaussian distribution quantified.

Applicable to both independent and correlated matrix ensembles.

Abstract

We investigate the linear statistics of random matrices with purely imaginary Bernoulli entries of the form $H_{pq} = \overline{H}_{qp} = \pm i$, that are either independently distributed or exhibit global correlations imposed by the condition $\sum_{q} H_{pq} = 0$. These are related to ensembles of so-called random tournaments and random regular tournaments respectively. Specifically, we construct a random walk within the space of matrices and show that the induced motion of the first $k$ traces in a Chebyshev basis converges to a suitable Ornstein-Uhlenbeck process. Coupling this with Stein's method allows us to compute the rate of convergence to a Gaussian distribution in the limit of large matrix dimension.