HAL : hal-00369621, version 4

arXiv : 0903.3528

DOI : 10.1214/10-AOP587

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Annals of Probability 39, 4 (2011) 1544-1590

Versions disponibles :

v1 (20-03-2009)

v2 (14-04-2009)

v3 (09-06-2010)

v4 (11-06-2010)

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

Charles Bordenave 1, Pietro Caputo 2, Djalil Chafai (  Auteur correspondant ) 3

(01/08/2011)

We consider the random reversible Markov kernel K obtained by assigning i.i.d. non negative 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, with alpha in (0,2). When 1<= \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_n K coincides with the one of the random symmetric matrix of the un-normalized weights (Levy 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 non trivial law wmu_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 wmu_alpha. We also study the limiting behavior of the invariant probability measure of K.

1 :	Institut de Mathématiques de Toulouse (IMT)
	Université Paul Sabatier - Toulouse III – Université Toulouse le Mirail - Toulouse II – Université des Sciences Sociales - Toulouse I – Institut National des Sciences Appliquées de Toulouse – CNRS : UMR5219
2 :	Dipartimento di Matematica [Roma TRE]
	Università degli Studi Roma TRE
3 :	Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
	Université Paris Est Marne-la-Vallée – Université Paris XII - Paris Est Créteil Val-de-Marne – CNRS : UMR8050 – Fédération de Recherche Bézout

Domaine

:

Mathématiques/Probabilités Mathématiques/Théorie spectrale

Mots Clés : Spectral theory – Objective method – Operator convergence – Stochastic matrices – Random matrices – Reversible Markov chains – Random walks – Random Graphs – Probability on trees – Random media – Heavy tailed distributions – alpha-stable laws – Poisson--Dirichlet laws – Point processes – Eigenvectors

hal-00369621, version 4

http://hal.archives-ouvertes.fr/hal-00369621

oai:hal.archives-ouvertes.fr:hal-00369621

Contributeur : Djalil Chafai <>

Soumis le : Jeudi 10 Juin 2010, 12:08:34

Dernière modification le : Jeudi 15 Septembre 2011, 21:06:53