The Dirichlet Markov Ensemble

We equip the polytope of n x n Markov matrices with the normalized trace of the Lebesgue measure of R-n2. 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 M is such a random matrix, then the empirical distribution built from the singular values of root n M tends as n -> infinity 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 root n M tends as n -> infinity 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/root n when n is large. (C) 2009 Elsevier Inc. All rights reserved.