The Dirichlet Markov Ensemble

We equip the polytope of n × 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 , … , 1 / n ) . We show that if M is such a random matrix, then the empirical distribution built from the singular values of n M tends as 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 n M tends as 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 n is large.