On the principal eigenvectors of random Markov matrices

TL;DR

This paper studies the invariant distributions of random Markov matrices derived from weighted complete digraphs, showing convergence to simple distributions under broad conditions and addressing open questions about their asymptotic behavior.

Contribution

It provides new results on the convergence of invariant distributions of random Markov matrices and answers open questions about their asymptotic uniformity.

Findings

Invariant distribution converges to inverse vertex weights under certain conditions.

Asymptotic uniformity of invariant distribution when edge weights have finite second moment.

Results hold even with heavy-tailed vertex weights.

Abstract

We analyze the invariant distributions of continuous-time and discrete-time random walks on randomly weighted complete digraphs. These distributions correspond to the principal left eigenvectors of the associated random Markov generators and kernels, viewed as random matrices. While much is known about the spectra of these matrices, relatively little is known about the principal left eigenvectors, which are delicate random objects for which no explicit form is known. We consider a broad class of such matrices obtained by associating random weights to the vertices and edges of the complete digraph. Our main result concerns the total variation distance between the invariant distribution of the continuous-time random walk and the distribution that is inversely proportional to the vertex weights. It states that, if the edge weights are i.i.d. with a finite $p$-th moment for some $p>4$, then this distance a.s. converges to zero as the number of vertices grows large, even when the vertex weights are heavy-tailed. We further answer a question of Bordenave, Caputo, and Chafa\"i by showing that, despite the dependence of the entries in the corresponding Markov kernel, its invariant distribution is asymptotically uniform a.s., so long as the edge weights have a finite second moment.