The Dirichlet Markov Ensemble

Djalil Chafai (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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 $(1/n,\ldots,1/n)$. 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.

Mots clés: Random matrices; Markov matrices; Dirichlet distributions; Spectral gap; singular values

Domaines: Probabilités [math.PR]; Théorie spectrale [math.SP]

Djalil Chafai. The Dirichlet Markov Ensemble. Journal of Multivariate Analysis, 2010, 101, pp.555-567. ⟨10.1016/j.jmva.2009.10.013⟩. ⟨hal-00175643v3⟩