We 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 <1, we show that the empirical spectral distribution of K converges without rescaling to a non trivial law wmu_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 wmu_alpha. We also study the limiting behavior of the invariant probability measure of K.