The Annals of Probability

Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph

Charles Bordenave, Pietro Caputo, and Djalil Chafaï

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Abstract

We consider the random reversible Markov kernel K obtained by assigning i.i.d. nonnegative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an α-stable law, α ∈ (0, 2). When 1 ≤ α < 2, we show that for a suitable regularly varying sequence κn of index 1 − 1/α, the limiting spectral distribution μα of κnK coincides with the one of the random symmetric matrix of the un-normalized weights (Lévy matrix with i.i.d. entries). In contrast, when 0 < α < 1, we show that the empirical spectral distribution of K converges without rescaling to a nontrivial law μ̃α supported on [−1, 1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of μα and μ̃α. We also study the limiting behavior of the invariant probability measure of K.

Article information

Source
Ann. Probab., Volume 39, Number 4 (2011), 1544-1590.

Dates
First available in Project Euclid: 5 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1312555808

Digital Object Identifier
doi:10.1214/10-AOP587

Mathematical Reviews number (MathSciNet)
MR2857250

Zentralblatt MATH identifier
1245.60008

Subjects
Primary: 47A10: Spectrum, resolvent 15A52 60K37: Processes in random environments 05C80: Random graphs [See also 60B20]

Keywords
Spectral theory objective method operator convergence stochastic matrices random matrices reversible Markov chains random walks random graphs probability on trees random media heavy-tailed distributions α-stable laws Poisson–Dirichlet laws point processes eigenvectors

Citation

Bordenave, Charles; Caputo, Pietro; Chafaï, Djalil. Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph. Ann. Probab. 39 (2011), no. 4, 1544--1590. doi:10.1214/10-AOP587. https://projecteuclid.org/euclid.aop/1312555808


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