  We equip the polytope of $n\times n$ Markov matrices with the normalized
trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. 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 $\bM$ is such a random
matrix, then the empirical distribution built from the singular values
of$\sqrt{n} \bM$ tends as $n\to\infty$ 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 $\sqrt{n} \bM$ tends as $n\to\infty$ to
the uniform distribution on the unit disc of the complex plane, and that
moreover, the spectral gap of $\bM$ is of order $1-1/\sqrt{n}$ when $n$ is
large.
