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Lipschitz continuity of the time constant for continuum percolation
Abstract
We consider the Boolean model of continuum percolation, where points are placed in ℝd by a Poisson point process and pairs of points with distance at most 1 are connected by an edge. The time constant is the limiting ratio of the chemical distance (i.e. graph distance) to the Euclidean distance for pairs of distant connected points. Yao, Chen, and Guo established the existence of a time constant in the supercritical regime. We show that above the critical intensity, the time constant is a Lipschitz continuous function of the intensity. The proof adapts a recent argument of Can, Nakajima, and Nguyen to the continuous setting.
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BibTeX
@misc{dubin2026lipschitz,
title = {{Lipschitz continuity of the time constant for continuum percolation}},
author = {Karoline Dubin and Christian Gorski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.31568},
eprint = {2605.31568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.31568},
}
RIS
TY - UNPB
TI - Lipschitz continuity of the time constant for continuum percolation
AU - Karoline Dubin
AU - Christian Gorski
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2605.31568
UR - https://arxiv.org/abs/2605.31568
AB - We consider the Boolean model of continuum percolation, where points are placed in ℝ^d by a Poisson point process and pairs of points with distance at most 1 are connected by an edge. The time constant is the limiting ratio of the chemical distance (i.e. graph distance) to the Euclidean distance for pairs of distant connected points. Yao, Chen, and Guo established the existence of a time constant in the supercritical regime. We show that above the critical intensity, the time constant is a Lipschitz continuous function of the intensity. The proof adapts a recent argument of Can, Nakajima, and Nguyen to the continuous setting.
ER -