Circular Law Theorem for Random Markov Matrices - Archive ouverte HAL Access content directly
Journal Articles Probability Theory and Related Fields Year : 2012

Circular Law Theorem for Random Markov Matrices


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.
Fichier principal
Vignette du fichier
cirmar.pdf (658.42 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-00310528 , version 1 (10-08-2008)
hal-00310528 , version 2 (11-08-2008)
hal-00310528 , version 3 (11-08-2008)
hal-00310528 , version 4 (11-08-2008)
hal-00310528 , version 5 (08-06-2010)



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



Gmail Mastodon Facebook X LinkedIn More