Circular Law Theorem for Random Markov Matrices

TL;DR

This paper proves that the eigenvalue distribution of scaled random Markov matrices converges to a uniform distribution on a disk, extending the circular law to a new class of stochastic matrices.

Contribution

It establishes a circular law for the spectrum of scaled random Markov matrices with i.i.d. entries, including the Dirichlet ensemble, under minimal assumptions.

Findings

Eigenvalue distribution converges to uniform law on a disk

Results apply to Dirichlet Markov ensembles

Convergence holds with probability one

Abstract

Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded density, mean m, and finite positive variance sigma^2. Let M be the nxn random Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its sum. In particular, when X11 follows an exponential law, then M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Our main result states that with probability one, the counting probability measure of the complex spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law on the centered disk of radius sigma/m. The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.