We consider the random reversible Markov kernel K on the complete graph with n vertices obtained by putting i.i.d. positive weights of law L on the n(n+1)/2 edges of the graph and normalizing each weight by the corresponding row sum. We have already shown in a previous work that if L has finite second moment then, as n goes to infinity, the limiting spectral distribution of n^{1/2}K is Wigner's semi-circle law. In the present work, we consider the case where L belongs to the domain of attraction of an alpha-stable law, 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 of kappa_n K coincides with the one of the random symmetric matrix of the un-normalized weights (i.i.d. entries). This result is extended to the marginal case alpha=1 under a mild extra assumption. In contrast, when 0 < alpha < 1, we show that the empirical spectral distribution of K converges, without any rescaling, to a non-trivial law supported on [-1,1], whose moments are the return probabilities of the random walk on a suitable Poisson weighted infinite tree of Aldous. The limiting operator is naturally linked with the Poisson-Dirichlet distribution PD(alpha,0).