The Dirichlet Markov Ensemble

We equip the polytope of n × n Markov matrices with the normalized trace of the Lebesgue measure of Rn 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 ifM is such a randommatrix, then the empirical distribution built from the singular values of √ nM 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 √ nM 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. AMS 2000 Mathematical Subject Classification: 15A52; 15A51; 15A42; 60F15; 62H99.