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

We study visibility inside the vacant set of three models in Rᵈ with slow decay of spatial correlations: Brownian interlacements, Poisson cylinders and Poisson-Boolean models. Let Qₓ 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 $0$, Qₓ/δ\|x\| converges weakly, as x→∞, to the exponential distribution with an explicit intensity, which depends on the parameters of the respective model. The scaling function δᵣ is the visibility window introduced in arXiv:2304.10298, a length scale of correlations in the visible set at distance r from $0$.