Circular Law Theorem for Random Markov Matrices

Djalil Chafai (1, 2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 UPTE - Physiopathologie et Toxicologie Expérimentales

Let X be a random matrix of dimension n with i.i.d. non negative real entries with unit mean, finite positive variance sigma^2 and finite fourth moment. Let M be the random Markov matrix with i.i.d. rows obtained by dividing each row of X by its 1-norm. It belongs to the Dirichlet Markov Ensemble when X has exponential entries. We show that with probability one, the empirical spectral distribution of n^{1/2}M converges weakly as n goes to infinity to the uniform law on the centered disc of radius sigma. Moreover, the spectral gap of M is of order n^{-1/2}. We believe that the finite fourth moment assumption is technical and might be removed.

Mots clés: Random matrices; Spectral gap; Singular values; Eigenvalues; Circular law; Rank-one deformations; Non-normal operators; Logarithmic potential; Markov matrices

Domaines: Probabilités [math.PR]; Théorie spectrale [math.SP]

Djalil Chafai. Circular Law Theorem for Random Markov Matrices. 2008. ⟨hal-00310528v1⟩