Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph

TL;DR

This paper analyzes the spectral properties of large random reversible Markov chains with heavy-tailed weights, revealing different limiting behaviors depending on the tail index, and connects these to stable laws and infinite tree structures.

Contribution

It characterizes the spectral distributions of heavy-tailed weighted Markov chains, linking them to stable laws and Poisson weighted infinite trees, and studies their invariant measures.

Findings

For 1 ≤ α < 2, spectral distribution matches that of un-normalized weights.

For 0 < α < 1, spectral distribution converges to a law supported on [-1,1].

The invariant measure's limiting behavior is also characterized.

Abstract

We consider the random reversible Markov kernel K obtained by assigning i.i.d. nonnegative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an $\alpha$-stable law, $\alpha\in(0,2)$. When $1\leq\alpha<2$, we show that for a suitable regularly varying sequence $\kappa_n$ of index $1-1/\alpha$, the limiting spectral distribution $\mu_{\alpha}$ of $\kappa_nK$ coincides with the one of the random symmetric matrix of the un-normalized weights (L\'{e}vy matrix with i.i.d. entries). In contrast, when $0<\alpha<1$, we show that the empirical spectral distribution of K converges without rescaling to a nontrivial law $\widetilde{\mu}_{\alpha}$ supported on [-1,1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of $\mu_{\alpha}$ and $\widetilde{\mu}_{\alpha}$. We also study the limiting behavior of the invariant probability measure of K.