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

+ self-adjointness, + almost-sure convergence, + stylistic changes +fixup in abstract and bibliographyInternational audienceWe consider the random reversible Markov kernel K obtained by assigning i.i.d. non negative 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, with alpha in (0,2). When 1<= \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_n K coincides with the one of the random symmetric matrix of the un-normalized weights (Levy matrix with i.i.d. entries). In contrast, when 0< alpha