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

Spectrum of large random Markov chains: heavy-tailed weights on the oriented complete graph