On the critical value of void percolation in high dimensions

Abstract

In the random Lorentz gas, a point tracer moves through spherical obstacles of (reduced) density $Φ$ centered on a Poisson point process in $d$-dimensional space, $\mathbb{R}^d$, and becomes localized once the vacant space stops percolating, at $Φ_{\rm p}(d)$. The dimensional scaling of this void percolation threshold has remained elusive. Prior bounds, conjectural ones included, ranged from $Ω(1)$ to $O(d\ln d)$, leaving even the scaling undetermined. We here prove that $Φ_{\rm p}(d)=Θ(d)$, and in particular that $\tfrac12\,d \le Φ_{\rm p}(d) \le 5\,d$ for all sufficiently large $d$. The upper bound counts hierarchical families of escape waypoints called tree embeddings; the lower bound is via a weighted second-moment estimate on an explicit ensemble of ballistic escape routes. A linear scaling of $Φ_{\rm p}(d)$ excludes the $d\ln d$ growth suggested by analogy with other systems in the random first-order transition universality class and is consistent with the numerical estimate $Φ_{\rm p}/d \to 3.40(5)$. Because the mean-field dynamical transition is likewise linear, $Φ_{\rm d}\simeq 2.40\,d$, our result fixes the activated window $[Φ_{\rm d},Φ_{\rm p}]$ as a bounded---rather than diverging---correction to mean-field dynamical arrest in sufficiently high $d$.