On the visibility window for Brownian interlacements, Poisson cylinders and Boolean models

TL;DR

This paper investigates the asymptotic behavior of the largest visible ball in the vacant set of Brownian interlacements, Poisson cylinders, and Boolean models, revealing a universal exponential limit after appropriate scaling.

Contribution

It establishes the weak convergence of the scaled maximum visible radius conditioned on visibility, introducing a new understanding of the visibility window in these models.

Findings

The scaled maximum visible radius converges to an exponential distribution.

The explicit intensity of the exponential distribution depends on model parameters.

The results apply to models with slow decay of spatial correlations.

Abstract

We study visibility inside the vacant set of three models in $\mathbb R^d$ with slow decay of spatial correlations: Brownian interlacements, Poisson cylinders and Poisson-Boolean models. Let $Q_x$ be the radius of the largest ball centered at $x$ every point of which is visible from $0$ through the vacant set of one of these models. We prove that conditioned on $x$ being visible from $0$, $Q_x/\delta_{\|x\|}$ converges weakly, as $x\to\infty$, to the exponential distribution with an explicit intensity, which depends on the parameters of the respective model. The scaling function $\delta_r$ is the visibility window introduced in arXiv:2304.10298, a length scale of correlations in the visible set at distance $r$ from $0$.