  We consider the random Markov matrix obtained by assigning i.i.d.
non-negative weights to each edge of the complete oriented graph. In this
study, the weights have unbounded first moment and belong to the domain of
attraction of an alpha-stable law. We prove that as the dimension tends to
infinity, the empirical measure of the singular values tends to a probability
measure which depends only on alpha, characterized as the expected value of the
spectral measure at the root of a weighted random tree. The latter is a
generalized two-stage version of the Poisson weighted infinite tree (PWIT)
introduced by David Aldous. Under an additional smoothness assumption, we show
that the empirical measure of the eigenvalues tends to a non-degenerate
isotropic probability measure depending only on alpha and supported on the unit
disc of the complex plane. We conjecture that the limiting support is actually
formed by a strictly smaller disc.
