Circular Law Theorem for Random Markov Matrices

Details Circular Law Theorem for Random Markov Matrices Circular Law Theorem for Random Markov Matrices

2008-08-11

arxiv.org/abs/0808.1502v3

Probability Theory and Related Fields 152, 3-4 (2012) 751-779

Mathematics

Probability

technical update via http://HAL.archives-ouvertes.fr

Scientific paper

10.1007/s00440-010-0336-1

Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded density, mean m, and finite positive variance sigma^2. Let M be the nxn random Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its sum. In particular, when X11 follows an exponential law, then M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Our main result states that with probability one, the counting probability measure of the complex spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law on the centered disk of radius sigma/m. The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.

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