In this paper, we recall the existence 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 Følner function $exp\left(n^{\frac{2\alpha }{1-\alpha }}\right)$. This result was proved by Erschler (see [4], [3]); we give a more detailed proof of this construction in the appendix. 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...