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On s’intéresse à une marche aléatoire simple sur un amas infini issu d’un processus de percolation surcritique sur les arêtes de ${\mathbb{Z}}^d(d\ge 2)$ de loi $Q$. On montre que la transformée de Laplace du nombre de points visités au temps $n$, noté $N_n$, a un comportement similaire au cas où la marche évolue dans ${\mathbb{Z}}^d$. Plus précisément, on établit que pour tout $0\<\alpha \<1$, il existe des constantes $C_i$, $C_s\>0$ telles que pour presque toute réalisation de la percolation telle que l’origine appartienne à l’amas infini et pour $n$ assez grand, ...

In this paper, we recall the existence of graphs with bounded valency such that the simple random walk has a return probability at time $n$ at the origin of order $exp(-n^{\alpha }),$ for fixed $\alpha \in [0,1[$ and with Følner function $exp\left(n^{\frac{2\alpha }{1-\alpha }}\right)$. This result was proved by Erschler (see [4], [3]); we give a more detailed proof of this construction in the appendix. In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random walk on...

A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalized version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models.