  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.
