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arXiv · 0704.2337

Existence of graphs with sub exponential transitions probability decay and applications

Clement Rau
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Abstract

In this paper, we present a complete proof of the construction of graphs with bounded valency such that the simple random walk has a return probability at time $n$ at the origin of order $exp(-n^α),$ for fixed $α\in [0,1[$ and with Folner function $exp(n^{\frac{2α}{1-α}})$. We begin by giving a more detailled proof of this result contained in (see \cite{ershdur}). 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 an infinite cluster on the percolation model.

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Clement Rau. 2007-04-18. Existence of graphs with sub exponential transitions probability decay and applications. https://arxiv.org/abs/0704.2337

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