On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements

TL;DR

This paper investigates the connectivity of the vacant set of Wiener sausages and Brownian interlacements in high dimensions, proving uniqueness of the infinite component and providing bounds on multiple components in finite ensembles.

Contribution

It establishes the almost sure uniqueness of the infinite component in the vacant set of Brownian interlacements and derives sharp probability bounds for multiple components in finite Wiener sausage ensembles.

Findings

The vacant set of Brownian interlacements has at most one infinite connected component almost surely.

Sharp polynomial bounds are provided for the probability of multiple components in finite Wiener sausage ensembles.

The main proof involves bounds on joint visitation probabilities of Brownian motions to hemiballs.

Abstract

We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in $\mathbb R^d$ in the transient dimensions $d \geq 3$. We prove that the vacant set of Brownian interlacements contains at most one infinite connected component almost surely. For finite ensembles of Wiener sausages, we provide sharp polynomial bounds on the probability that their vacant set contains at least $2$ connected components in microscopic balls. The main proof ingredient is a sharp polynomial bound on the probability that several Brownian motions visit jointly all hemiballs of the unit ball while avoiding a slightly smaller ball.