Percolation in the Boolean model with convex grains in high dimension

TL;DR

This paper studies how percolation occurs in high-dimensional Boolean models with convex grains, analyzing the asymptotic behavior of percolation probability and thresholds for different configurations.

Contribution

It provides the first asymptotic analysis of percolation thresholds in high-dimensional Boolean models with convex grains, considering two different translation settings.

Findings

Asymptotic behavior of percolation probability characterized

Percolation threshold estimates derived for high dimensions

Results applicable to models with convex, symmetric grains

Abstract

We investigate percolation in the Boolean model with convex grains in high dimension. For each dimension d, one fixes a compact, convex and symmetric set K $\subset$ R d with non empty interior. In a first setting, the Boolean model is a reunion of translates of K. In a second setting, the Boolean model is a reunion of translates of K or $\rho$K for a further parameter $\rho$ $\in$ (1, 2). We give the asymptotic behavior of the percolation probability and of the percolation threshold in the two settings.