Subcritical sharpness for multiscale Boolean percolation

We consider a multiscale Boolean percolation on ${\mathbb{R}}^d$ with radius distribution $\mu$ on $[1,+\mathrm{\infty })$, $d\ge 2$. 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 $\mu {\mathrm{\mid }}_{[1,\kappa ]}$ 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 $1$).