Asymptotics of the critical time in Wiener sausage percolation with a small radius

Abstract

We consider a continuum percolation model on ${\mathbb{R}}^d$, where $d\ge 4$.The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whoseinitial points are distributed according to a homogeneous Poisson point process.It was established in a previous work by Erhard, Martínez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t\_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of the critical time when the radius $r$ converges to $0$. The latter does not seem to be deducible from simple scaling arguments. We prove that for $d\ge 4$, there is a positive constant $c$ such that$c^{-1}\sqrt{\log (1/r)}\le t\_c(r)\le c\sqrt{\log (1/r)}$ when $d=4$ and $c^{-1}{r}^{(4-d)/2}\le t\_c(r)\le c{r}^{(4-d)/2}$ when $d\ge 5$, as $r$ converges to $0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest.