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

TL;DR

This paper analyzes the asymptotic behavior of the critical time for percolation in a Wiener sausage model on ^d with small radius, establishing bounds for the critical time as the radius approaches zero.

Contribution

It provides the first precise asymptotic bounds for the critical percolation time as the Wiener sausage radius tends to zero in dimensions four and higher.

Findings

Critical time scales as (1/r)^{1/2} in 4D.

Critical time scales as r^{(4-d)/2} in dimensions d  5 and above.

Derived moment estimates on Wiener sausage capacity.

Abstract

We consider a continuum percolation model on $\R^d$, where $d\geq 4$.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 $t\_c(r)$ 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 $d\geq 4$, there is a positive constant $c$ such that$c^{-1}\sqrt{\log(1/r)}\leq t\_c(r)\leq c\sqrt{\log(1/r)}$ when $d=4$ and $c^{-1}r^{(4-d)/2}\leq t\_c(r) \leq c\ r^{(4-d)/2}$ when $d\geq 5$, as $r$ converges to $0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest.