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, ℝd, and becomes localized once the vacant space stops percolating, at Φp(d). The dimensional scaling of this void percolation threshold has remained elusive. Prior bounds, conjectural ones included, ranged from Ω(1) to O(dln d), leaving even the scaling undetermined. We here prove that Φp(d)=Θ(d), and in particular that 1/2 d ≤ Φp(d) ≤ 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 Φp(d) excludes the dln d growth suggested by analogy with other systems in the random first-order transition universality class and is consistent with the numerical estimate Φp/d → 3.40(5). Because the mean-field dynamical transition is likewise linear, Φd≃ 2.40 d, our result fixes the activated window [Φd,Φp] as a bounded---rather than diverging---correction to mean-field dynamical arrest in sufficiently high d.
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BibTeX
@misc{charbonneau2026critical,
title = {{On the critical value of void percolation in high dimensions}},
author = {Patrick Charbonneau and Marcus Michelen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07196},
eprint = {2611.07196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07196},
}
RIS
TY - UNPB
TI - On the critical value of void percolation in high dimensions
AU - Patrick Charbonneau
AU - Marcus Michelen
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07196
UR - https://arxiv.org/abs/2611.07196
AB - 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, ℝ^d, and becomes localized once the vacant space stops percolating, at Φ_p(d). The dimensional scaling of this void percolation threshold has remained elusive. Prior bounds, conjectural ones included, ranged from Ω(1) to O(dln d), leaving even the scaling undetermined. We here prove that Φ_p(d)=Θ(d), and in particular that 1/2 d ≤ Φ_p(d) ≤ 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_p(d) excludes the dln d growth suggested by analogy with other systems in the random first-order transition universality class and is consistent with the numerical estimate Φ_p/d → 3.40(5). Because the mean-field dynamical transition is likewise linear, Φ_d≃ 2.40 d, our result fixes the activated window [Φ_d,Φ_p] as a bounded---rather than diverging---correction to mean-field dynamical arrest in sufficiently high d.
ER -