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.
"Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph." Ann. Probab. 39 (4) 1544 - 1590, July 2011. https://doi.org/10.1214/10-AOP587