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Marches aléatoires sur un amas infini de percolation.

Abstract : In this thesis, we consider a simple random walk on the infinite cluster of the percolation model on the edges of $\Z^d\ (d\geq 2)$ with law $Q$, in the surcritical case. We look at the Laplace transformation of some functional of local times of this walk. In the first part, we investigate the particular case of the Laplace transformation of
the number of visited sites up to time $n$, called $N_n$. We prove that this quantity has the same behaviour as the random walk on $\Z^d$.
More precisely, we show for all $0<\alpha<1$, there exists some constants $C_i, \ C_s >0$ such that for almost all realisations of
the percolation such that the origin belongs to the infinite cluster and for large enough $n$,$$ e^{-C_i n^{ \frac{d}{d+2} } } \leq \E_0^{\omega} ( \alpha^{N_n} )
\leq e^{-C_sn^{ \frac{d}{d+2} }}.$$
In the second part, we extend this kind of estimate
for other functionals. For these problems, the main work is to get the upper bound. Our approach is based, first on finding an isoperimetric inequality on the infinite cluster and secondly to lift it on a wreath
product, which enables us to get an upper bound of the return probability of a particular random walk on this wreath product. The introduction of a wreath product is motivated by the fact that the
return probability on such graph is linked to the Laplace transform of some functional of the locals times for a good choice of the fibers.
Finally, in the last part we explain with details and in a general case, following ideas of A.Erschler, how to get a isoperimetric inequality on a wreath product of two graphs from an isoperimetric inequality on each graphs.
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Submitted on : Thursday, October 19, 2006 - 6:48:08 PM
Last modification on : Wednesday, October 10, 2018 - 1:26:48 AM
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Clément Rau. Marches aléatoires sur un amas infini de percolation.. Mathématiques [math]. Université de Provence - Aix-Marseille I, 2006. Français. ⟨tel-00108175⟩

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