Circular law theorem for random Markov matrices

• 成果类型: Article
• 署名作者: Bordenave, Charles; Caputo, Pietro; Chafai, Djalil
• 署名单位: Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Federale Toulouse Midi-Pyrenees (ComUE); Institut National des Sciences Appliquees de Toulouse; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Roma Tre University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0336-1
• 发表日期: 2012
• 页码: 751-779
• 关键词: singular-values; eigenvalues; BEHAVIOR

摘要

Let (X-jk)(jk >= 1) be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance sigma(2). Let M be the n x n random Markov matrix with i.i.d. rows defined by M-jk = X-jk/( (j)1+...+X-jn). In particular, when X-11 follows an exponential law, the random matrix M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Let lambda(1),...,lambda(n) be the eigenvalues of root nM i.e. the roots in C of its characteristic polynomial. Our main result states that with probability one, the counting probability measure 1/n delta lambda(1)+...+1/n delta(lambda n) converges weakly as n -> infinity to the uniform law on the disk {z is an element of C : vertical bar z vertical bar <= m(-1)sigma}. The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.