# Marches Aléatoires avec Conductances Aléatoires

Abstract : This thesis deals with an important class of RWRE called random walks among random conductances. We give tree principal results showing opposite behaviors, anomalous and standard, of the heat-kernel of random walks among polynomial lower tail random conductances. The first two results (cf. Chapter 2) concern discrete-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances with values in the interval $[0,1]$, with polynomial tail near $0$ with exponent $\gamma>0$. We first prove for all $d>4$ that the return probability shows an anomalous decay (non-gaussian) that approaches (up to sub-polynomial terms) a random constant times $1/n^{2}$ when we push the power $\gamma$ to zero. In contrast, we prove that the heat-kernel decay is, in a logarithmic sense, as close as we want to the standard decay $1/n^{d/2}$ for large values of the parameter $\gamma$. We consider in the third result (cf. Chapter 3) the same Markov chains in the continuous-time case and study the asymptotic behavior of the return probability. We show that for $\gamma> d/2$ the spectral dimension is standard, i.e. equal to $d$. As an expected consequence, the same result holds for the discrete-time case.
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https://tel.archives-ouvertes.fr/tel-00523660
Submitted on : Friday, January 7, 2011 - 10:49:14 AM
Last modification on : Wednesday, October 10, 2018 - 1:26:16 AM
Long-term archiving on: : Friday, December 2, 2016 - 12:54:41 PM

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• HAL Id : tel-00523660, version 2

### Citation

Omar Boukhadra. Marches Aléatoires avec Conductances Aléatoires. Mathématiques [math]. Université de Provence - Aix-Marseille I, 2010. Français. ⟨tel-00523660v2⟩

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