The Dirichlet Markov Ensemble
Résumé
We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of 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 $M$ is such a random matrix, then the empirical spectral distribution of $nMM^\top$ tends as $n\to\infty$ to a Marchenko-Pastur distribution. This phenomenon complements an already known result on the sub-dominant eigenvalue of certain random matrices with independent rows, which suggests that the typical spectral gap of a uniform random Markov matrix is of order $1-1/\sqrt{n}$ when $n$ is large. However, some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we conjecture that the empirical distribution of the complex spectrum of $\sqrt{n}M$ tends as $n\to\infty$ to the uniform distribution on the unit disc of the complex plane.
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