The internal diffusion limited aggregation (DLA) has been introduced by Diaconis and Fulton [Rend. Sem. Mat. Univ. Pol. Torino 49, 95-119 (1991)]. It is a growth model defined on an infinite set and associated to a Markov chain on this set. We focus here on sets which are finitely generated groups with exponential growth. We prove a shape theorem for the internal DLA on such groups associated to symmetric random walks. For that purpose, we introduce a new distance associated to the Green function, which happens to have some interesting properties. In the case of homogeneous trees, we also get the right order for the fluctuations of that model around its limiting shape.

We consider the autoregressive model on R d defined by the stochastic recursionThe critical case, when E[log A 1 ] = 0, was studied by Babillot, Bougerol and Elie, who proved that there exists a unique invariant Radon measure ν for the Markov chain {X n }. In the present paper we prove that the weak limit of properly dilated measure ν exists and defines a homogeneous measure on R d \ {0}.Résumé. Nous considérons le modèle autorégressif sur R d défini par récurrence par l'équation stochastique X n = A n X n−1 + B n , où {(B n , A n )} sont des variables aléatoires à valeurs dans R d × R + , indépendantes et de même loi. Le cas critique, c'est-à-dire lorsque E[log A 1 ] = 0, a été étudié par Babillot, Bougerol et Elie, qui ont montré qu'il existe une et une seule mesure de Radon ν invariante pour la chaîne de Markov {X n }. Dans ce papier nous démontrons que la mesure ν, convenablement dilatée, converge faiblement vers une mesure homogène sur R d \ {0}.

Abstract.We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel-Leader graph DL(q, r), where q, r ≥ 2. The latter is the horocyclic product of two homogeneous trees with respective degrees q + 1 and r + 1. When q = r, it is the Cayley graph of the wreath product (lamplighter group) Z q ≀ Z with respect to a natural set of generators. We describe the full Martin compactification of these random walks on DL-graphs and, in particular, lamplighter groups. This completes and provides a better approach to previous results of Woess, who has determined all minimal positive harmonic functions.