Spectrum of large random Markov chains: heavy-tailed weights on the oriented complete graph

TL;DR

This paper studies the spectral properties of large random Markov chains with heavy-tailed weights on complete directed graphs, revealing limiting spectral measures influenced by stable laws and tree structures.

Contribution

It introduces a new analysis of spectral measures for Markov matrices with heavy-tailed weights, connecting them to stable laws and generalized Poisson weighted infinite trees.

Findings

Empirical singular value distribution converges to a measure depending on alpha.

Eigenvalue distribution tends to a non-degenerate isotropic measure on the unit disc.

Limiting spectral support is conjectured to be a smaller disc than the unit circle.

Abstract

We consider the random Markov matrix obtained by assigning i.i.d. non-negative weights to each edge of the complete oriented graph. In this study, the weights have unbounded first moment and belong to the domain of attraction of an alpha-stable law. We prove that as the dimension tends to infinity, the empirical measure of the singular values tends to a probability measure which depends only on alpha, characterized as the expected value of the spectral measure at the root of a weighted random tree. The latter is a generalized two-stage version of the Poisson weighted infinite tree (PWIT) introduced by David Aldous. Under an additional smoothness assumption, we show that the empirical measure of the eigenvalues tends to a non-degenerate isotropic probability measure depending only on alpha and supported on the unit disc of the complex plane. We conjecture that the limiting support is actually formed by a strictly smaller disc.