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arXiv · 2607.29536

Multi-hop visibility through the vacant set of Poissonian obstacles

Yingxin Mu·Artem Sapozhnikov
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Abstract

We study multi-hop visibility inside the vacant set of three obstacle models in $\mathbb R^d$ with slow decay of spatial correlations and disparate obstacle geometries: Poisson-Boolean models with general i.i.d. radii distributions, Poisson cylinders and Brownian interlacements. For any $N\geq 0$, we obtain sharp bounds on the probability $P_{\mathrm{vis}}^N(r)$ of visibility to distance $r$ via $(N+1)$ hops in terms of the (explicit) probability of direct visibility to distance $r$ in a given direction, generalizing our earlier result from arXiv:2304.10298 for $N=0$. We observe a universal behavior of $P_{\mathrm{vis}}^N(r)$ in terms of two characteristic scales. In the three models of interest, these scales are generally the same, but with some anomalous exceptions in low dimensions.

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Yingxin Mu, Artem Sapozhnikov. 2026-07-31. Multi-hop visibility through the vacant set of Poissonian obstacles. https://arxiv.org/abs/2607.29536

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