The Dirichlet Markov Ensemble

We equip the polytope of n×nn×n Markov matrices with the normalized trace of the Lebesgue measure of Rn2Rn2. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n,…,1/n)(1/n,…,1/n). We show that if M is such a random matrix, then the empirical distribution built from the singular values of nM tends as n→∞n→∞ 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 nM tends as n→∞n→∞ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of M is of order 1−1/n when nn is large.