Circular Law Theorem for Random Markov Matrices

Abstract : 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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Charles Bordenave, Pietro Caputo, Djalil Chafai. Circular Law Theorem for Random Markov Matrices. Probability Theory and Related Fields, Springer Verlag, 2012, 152 (3-4), pp.751-779. ⟨10.1007/s00440-010-0336-1⟩. ⟨hal-00310528v5⟩

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