Spectrum of large random reversible Markov chains

Charles Bordenave (1) , Pietro Caputo (2) , Djalil Chafai (1, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Dipartimento di Matematica [Roma TRE]
3 UPTE - Physiopathologie et Toxicologie Expérimentales

In this work, we adopt a Random Matrix Theory point of view to study the spectrum of large reversible Markov chains in random environment. As the number of states tends to infinity, we consider both the almost sure global behavior of the spectrum, and the local behavior at the edge including the so called spectral gap. We study presently two simple models. The first one is on the complete graph while the second is on the chain graph (birth-and-death dynamics). These two models exhibit different scalings and limiting objects. The first model is related to the semi--circle law and Wigner's theorem. It contains as a special case a natural reversible Dirichlet Markov Ensemble. The second model is related to homogenization and also to asymptotics for the roots of random orthogonal polynomials. A special case gives rise to the arc--sine law as in a theorem by Erdos & Turan. This work raises several open problems.

Mots clés: tridiagonal matrices; birth-and-death processes; spectral analysis; orthogonal polynomials; weak and strong limit theorems; homogenization.; homogenization; random matrices; random graphs; reversible Markov chains; random walks; random environment; spectral gap; Wigner semi--circle law; arc--sine law

Domaines: Probabilités [math.PR]

Charles Bordenave, Pietro Caputo, Djalil Chafai. Spectrum of large random reversible Markov chains. 2008. ⟨hal-00337409v1⟩