  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 assumed to be in
the domain of attraction of an $\alpha$-stable law, $\alpha\in(0,2)$. When
$1\leq\alpha<2$, we show that for a suitable regularly varying sequence
$\kappa_n$ of index $1-1/\alpha$, the limiting spectral distribution
$\mu_{\alpha}$ of $\kappa_nK$ coincides with the one of the random symmetric
matrix of the un-normalized weights (L\'{e}vy 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 nontrivial law $\widetilde{\mu}_{\alpha}$
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
$\widetilde{\mu}_{\alpha}$. We also study the limiting behavior of the
invariant probability measure of K.
