Existence of graphs with sub exponential transitions probability decay and applications

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.