Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^d$

TL;DR

This paper investigates the sharpness of the phase transition in continuum percolation on  space, establishing exponential tail bounds for cluster sizes and providing new lower bounds for the critical intensity, supported by theoretical and simulation results.

Contribution

It introduces a new approach to analyze cluster tail behavior and derives improved lower bounds for the critical intensity in continuum percolation models.

Findings

Exponential tail bounds for cluster volume and diameter when grains are bounded

New lower bounds for the critical intensity, validated by simulations

Bounds are close to the true critical values with confidence intervals

Abstract

We consider the Boolean model $Z$ on $\mathbb{R}^d$ with random compact grains, i.e. $Z := \bigcup_{i \in \mathbb{N}} (X_i + Z_i)$ where $\eta_t := \{X_1, X_2, \dots\}$ is a Poisson point process of intensity $t$ and $(Z_1, Z_2, \dots)$ is an i.i.d. sequence of compact grains (not necessarily balls). We will show, that the volume and diameter of the cluster of a typical grain in $Z$ have an exponential tail if the diameter of the typical grain is a.s. bounded by some constant. To achieve this we adapt the arguments of \cite{duminil2015newproof} and apply a new construction of the cluster of the typical grain together with arguments related to branching processes. In the second part of the paper, we obtain new lower bounds for the boolean model with deterministic grains. Some of these bounds are rigorous, while others are obtained via simulation. The simulated bounds are very close to the "true" values and come with confidence intervals.