Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^d$

We consider the Boolean model $Z$ on $\mathbb{R}^d$ with random compact grains, i.e. $Z := \bigcup_{i \in \mathbb{N}} (X_i + Z_i)$ where $\eta_t := \{X_1, X_2, \dots\}$ is a Poisson point process of intensity $t$ and $(Z_1, Z_2, \dots)$ is an i.i.d. sequence of compact grains (not necessarily balls). We will show, that the volume and diameter of the cluster of a typical grain in $Z$ have an exponential tail if the diameter of the typical grain is a.s. bounded by some constant. To achieve this we adapt the arguments of \cite{duminil2015newproof} and apply a new construction of the cluster of the typical grain together with arguments related to branching processes. In the second part of the paper, we obtain new lower bounds for the boolean model with deterministic grains. Some of these bounds are rigorous, while others are obtained via simulation. The simulated bounds are very close to the "true" values and come with confidence intervals.