Subcritical sharpness for multiscale Boolean percolation

TL;DR

This paper studies a multiscale Boolean percolation model in high-dimensional space, demonstrating sharp phase transition behavior for large scale factors and showing that connectivity depends only on the fractal component of the model.

Contribution

It establishes the sharpness of the subcritical phase for multiscale Boolean percolation with large scale factors and clarifies the role of the fractal part in connectivity.

Findings

Subcritical phase is sharp for large scale factors.

Connectivity depends only on the fractal part of the model.

The model exhibits phase transition behavior similar to classical percolation.

Abstract

We consider a multiscale Boolean percolation on $\mathbb R^d$ with radius distribution $\mu$ on $[1,+\infty)$, $d\ge 2$. The model is defined by superposing the original Boolean percolation model with radius distribution $\mu$ with a countable number of scaled independent copies. The $n$-th copy is a Boolean percolation with radius distribution $\mu|_{[1,\kappa]}$ rescaled by $\kappa^{n}$. We prove that under some regularity assumption on $\mu$, the subcritical phase of the multiscale model is sharp for $\kappa $ large enough. Moreover, we prove that the existence of an unbounded connected component depends only on the fractal part (and not of the balls with radius larger than $1$).