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

Charles Bordenave (1) , Pietro Caputo (2) , Djalil Chafaï (3, 4) , Daniele Piras (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Dipartimento di Matematica [Roma TRE]
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 Université Paris Dauphine-PSL

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.

Mots clés: Spectral theory; Objective method; Operator convergence; Logarithmic potential; Random matrix; Random Graph; Heavy tailed distribution; Stable law

Domaines: Théorie spectrale [math.SP]; Probabilités [math.PR]; Combinatoire [math.CO]

Charles Bordenave, Pietro Caputo, Djalil Chafaï, Daniele Piras. Spectrum of large random Markov chains: heavy-tailed weights on the oriented complete graph. Random Matrices: Theory and Applications, 2017, 6 (2), pp.1750006. ⟨10.1142/S201032631750006X⟩. ⟨hal-01377076v2⟩