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

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 as- sumedtobeinthedomainofattractionofanα-stable law, α Σ (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 κn K 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