Matrix analysis
Eigenvalues and singular values of certain random matrices
Journal of Computational and Applied Mathematics
Journal of Multivariate Analysis
Distribution of Subdominant Eigenvalues of Matrices with Random Rows
SIAM Journal on Matrix Analysis and Applications
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We equip the polytope of nxn 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 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.