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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References in corpus (6)

• The circular law for random matrices
• Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem
• Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph
• Spectrum of non-Hermitian heavy tailed random matrices
• The Dirichlet Markov Ensemble
• Circular law for non-central random matrices

Cited by in corpus (22)

• Around the circular law
• Universality and the circular law for sparse random matrices
• Generating random quantum channels
• Spectrum of non-Hermitian heavy tailed random matrices
• Circular law for random matrices with exchangeable entries
• Random doubly stochastic matrices: The circular law
• A note on the Marchenko-Pastur law for a class of random matrices with dependent entries
• Spectrum of Markov generators on sparse random graphs
• Spectra of Random Stochastic Matrices and Relaxation in Complex Systems
• Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application
• Circular law for random matrices with unconditional log-concave distribution
• Breakdown of random matrix universality in Markov models
• Spectrum of large random Markov chains: heavy-tailed weights on the oriented complete graph
• From Boltzmann to random matrices and beyond
• Coulomb gases under constraint: some theoretical and numerical results
• Time-inhomogeneous random Markov chains
• Non-Hermitian random matrices with a variance profile (II): properties and examples
• Singular value distribution of dense random matrices with block Markovian dependence
• Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth
• Random stochastic matrices from classical compact Lie groups and symmetric spaces
• Statistics of the non-zero eigenvalues and singular values of low-rank random matrices with non-negative entries
• Entangled criticality and irreversibility in random Markov dynamics