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\|ₗ\| 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.
Mu et al. (Thu,) studied this question.