Percolation in a multiscale Boolean model

TL;DR

This paper studies percolation phenomena in a multiscale Boolean model, showing that for sufficiently large scale factors and below a critical rate, percolation does not occur, under certain integrability conditions.

Contribution

It establishes conditions under which multiscale Boolean models do not percolate for large scale factors, extending understanding of percolation thresholds in multiscale settings.

Findings

No percolation occurs for large enough scale factor when the Boolean model rate is below critical.

Proves results under optimal integrability assumptions.

Provides conditions for non-percolation in multiscale Boolean models.

Abstract

We consider percolation in a multiscale Boolean model. This model is defined as the union of scaled independent copies of a given Boolean model. The scale factor of the $n^{\textrm{th}}$ copy is $\rho^{-n}$. We prove, under optimal integrability assumptions, that no percolation occurs in the multiscale Boolean model for large enough $\rho$ if the rate of the Boolean model is below some critical value.