Percolation in a multiscale Boolean model

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 $n^{\text{th}}$ 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.