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 a stable law of index a. When 1< a <2, we show that for a suitable regularly varying sequence k_n of index 1 - 1/a, the limiting spectral distribution of k_n K coincides with the one of the random symmetric matrix of the un-normalized weights (i.i.d. entries). In contrast, when 0< a <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(a,0). The "critical" cases a=1 and a=2 are not solved here.