We equip the polytope of nxn Markov matrices with the normalized trace of the Lebesgue measure of . 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 is such a random matrix, then the empirical distribution built from the singular values of 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 tends as n-->[infinity] to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of is of order when n is large.
