Spectral statistics of Bernoulli matrix ensembles - a random walk approach (I)

TL;DR

This paper studies the eigenvalue statistics of Bernoulli matrices using a novel random walk approach, deriving a Fokker-Planck equation that links to Gaussian ensembles in the large matrix limit.

Contribution

It introduces a discrete random walk method to analyze Bernoulli matrix eigenvalues, connecting their statistics to Gaussian ensembles through a Fokker-Planck framework.

Findings

Eigenvalue distribution approximates Gaussian ensemble behavior for large matrices

Derived a Fokker-Planck equation describing the eigenvalue dynamics

Established the stationary distribution as a fixed trace Gaussian ensemble

Abstract

We investigate the eigenvalue statistics of random Bernoulli matrices, where the matrix elements are chosen independently from a binary set with equal probability. This is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues - an approach which is similar to Dyson's Brownian motion model but with important modifications. In particular, we show our process is described by a Fokker-Planck equation, up to an error margin which vanishes in the limit of large matrix dimension. The stationary solution of which corresponds to the joint probability density function of certain well-known fixed trace Gaussian ensembles.