The Dirichlet Markov Ensemble

TL;DR

This paper studies the spectral properties of random Markov matrices drawn from a Dirichlet distribution, revealing asymptotic behaviors of their singular values and complex spectra as matrix size grows.

Contribution

It introduces a new probabilistic model for random Markov matrices and analyzes their spectral distribution, including conjectures on the limiting spectral measure.

Findings

Singular values follow a Wigner quarter-circle distribution asymptotically.

Numerical simulations suggest the complex spectrum tends to a uniform distribution on the unit disk.

Spectral gap is approximately 1 - 1/√n for large n.

Abstract

We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean $(1/n,...,1/n)$. We show that if $\bM$ is such a random matrix, then the empirical distribution built from the singular values of$\sqrt{n} \bM$ tends as $n\to\infty$ to a Wigner quarter--circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of $\sqrt{n} \bM$ tends as $n\to\infty$ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of $\bM$ is of order $1-1/\sqrt{n}$ when $n$ is large.