Spectrum of Markov generators on sparse random graphs

TL;DR

This paper studies the spectral distribution of the generator matrix of a continuous-time random walk on sparse, randomly weighted directed graphs, revealing a Gaussian-deformed circular law as the limit.

Contribution

It establishes the convergence of the empirical spectral distribution for a broad class of sparse random graphs, including oriented Erdős-Rényi graphs, and characterizes the limiting distribution.

Findings

Convergence of spectral distribution to a Gaussian-deformed circular law.

Analysis of the invariant measure convergence to uniform distribution.

Estimates on extremal eigenvalues of the generator matrix.

Abstract

We investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if k<>j and L(j,j)=-sum(L(j,k),k<>j), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erd\"os-R\'enyi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.