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

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 $\alpha$-stable law, $\alpha\in(0,2)$. When $1\leq\alpha<2$, we show that for a suitable regularly varying sequence $\kappa_n$ of index $1-1/\alpha$, the limiting spectral distribution $\mu_{\alpha}$ of $\kappa_nK$ coincides with the one of the random symmetric matrix of the un-normalized weights (L\'{e}vy matrix with i.i.d. entries). In contrast, when $0<\alpha<1$, we show that the empirical spectral distribution of K converges without rescaling to a nontrivial law $\widetilde{\mu}_{\alpha}$ 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 $\mu_{\alpha}$ and $\widetilde{\mu}_{\alpha}$. We also study the limiting behavior of the invariant probability measure of K.