Visibility in Brownain interlacements, Poisson cylinders and Boolean models

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 Boolean model. For each of them, we obtain sharp asymptotic bounds on the probability of visibility to distance $r$ in some direction in terms of the probability of visibility to distance $r$ in a given direction. In dimensions $d\geq 4$, the ratio of the two probabilities has the same scaling $r^{2(d-1)}$ for all three models, but in lower dimensions the scalings are different. In particular, we improve some main results from arXiv:0905.4874 and arXiv:1709.09052.