Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster

TL;DR

This paper studies a continuum percolation model based on Brownian paths in ^d, establishing percolation phase transitions, the absence of percolation in one dimension, and the uniqueness of the unbounded cluster in higher dimensions.

Contribution

It introduces a novel continuum percolation model with Brownian paths, analyzing phase transitions and cluster uniqueness across different dimensions.

Findings

No percolation occurs in 1D for all times.

Percolation transition exists in dimensions 2 and 3.

Unbounded cluster is unique in the supercritical phase.

Abstract

We consider a continuum percolation model on $\R^d$, $d\geq 1$.For $t,\lambda\in (0,\infty)$ and $d\in\{1,2,3\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson point process with intensity $\lambda\textgreater{}0$.When $d\geq 4$, the Brownian paths are replaced by Wiener sausageswith radius $r\textgreater{}0$.We establish that, for $d=1$ and all choices of $t$, no percolation occurs,whereas for $d\geq 2$, there is a non-trivial percolation transitionin $t$, provided $\lambda$ and $r$ are chosen properly.The last statement means that $\lambda$ has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero(which is infinite when $d\in\{2,3\}$, but finite and dependent on $r$ when $d\geq 4$).We further show that for all $d\geq 2$, the unbounded cluster in the supercritical phase is unique.Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest.The present paper settles the basic properties of the model and should be viewed as a jumpboard for finer results.