Reçu le 23 avril 2006 ; révisée le 06 octobre 2006 ; accepté le 22 novembre 2006Résumé. Dans cet article nous démontrons un théorème de stabilité des probabilités de retour sur un groupe localement compact unimodulaire, séparable et compactement engendré. Nous démontrons que le comportement asymptotique de F * (2n) (e) ne dépend pas de la densité F sous des hypothèses naturelles. A titre d'exemple nouś etablissons que la probabilité de retour sur une large classe de groupes résolubles se comporte comme exp(−n 1/3 ).Abstract. The main result of this paper is a theorem about the stability of the return probability of symmetric random walks on a locally compact group which is unimodular, separable and compactly generated. We show that the asymptotic behavior of F * (2n) (e) does not depend on the choice of the density F under natural assumptions. As an example we show that the return probabilities for a large class of solvable groups of exponential volume growth, behaves like exp(−n 1/3 ).
MSC : Primary 60G50 ; secondary 22D25
We use the estimate of paths in Z 2 enclosing a null algebraic area to compute correction terms on the random walk on certain discrete Heisenberg groups. We obtain that the probability to return at the origin of the simple random walk on this group is 1 4n 2 +O( 1 n 3 ).
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