Existence of graphs with sub exponential transitions probability decay and applications

Details Existence of graphs with sub exponential transitions probability decay and applications Existence of graphs with sub exponential transitions probability decay and applications

2007-04-18

arxiv.org/abs/0704.2337v1

Mathematics

Probability

46 pages

Scientific paper

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 $exp(-n^{\alpha}),$ for fixed $\alpha \in [0,1[$ and with Folner function $exp(n^{\frac{2\alpha}{1-\alpha}})$. 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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