The Dirichlet Markov Ensemble

Details The Dirichlet Markov Ensemble The Dirichlet Markov Ensemble

2007-09-28

arxiv.org/abs/0709.4678v3

Journal of Multivariate Analysis 101, 555-567 (2010)

Mathematics

Probability

Improved version. Accepted for publication in JMVA

Scientific paper

10.1016/j.jmva.2009.10.013

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.

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