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

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, Mart'{i}nez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t\mathrm{\_}c(r)$ 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$c^{-1}\sqrt{\log(1/r)}\leq t_c®\leq c\sqrt{\log(1/r)}$when$d=4$and$c^{-1}r^{(4-d)/2}\leq t_c® \leq c
r^{(4-d)/2}$when$d\geq 5$,as$r$convergesto$0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest.