Visibility in the vacant set of the Brownian interlacements and the Brownian excursion process

TL;DR

This paper investigates the visibility properties within the vacant set of Brownian interlacements and excursions, revealing phase transitions and asymptotic behaviors in Euclidean space and the unit disc.

Contribution

It provides new estimates for visibility asymptotics in Brownian interlacements and characterizes phase transitions in the Brownian excursion process.

Findings

Visibility exhibits phase transition behavior.

Critical values for visibility are identified.

No visibility to infinity at the critical intensity.

Abstract

We consider the Brownian interlacements model in Euclidean space, introduced by A.S. Sznitman in \cite{sznitman2013scaling}. We give estimates for the asymptotics of the visibility in the vacant set. We also consider visibility inside the vacant set of the Brownian excursion process in the unit disc and show that it undergoes a phase transition regarding visibility to infinity as in \cite{benjamini2009visibility}. Additionally, we determine the critical value and that there is no visibility to infinity at the critical intensity.