Sharpness of phase transition for Voronoi percolation in hyperbolic space

In this paper, we consider Voronoi percolation in the hyperbolic space $\mathbb{H}^d$ ($d\ge 2$) and show that the phase transition is sharp. More precisely, we show that for Voronoi percolation with parameter $p$ generated by a homogeneous Poisson point process with intensity $\lambda$, there exists $p_c:=p_c(\lambda,d)$ such that the probability of a monochromatic path from the origin reaching a distance of $n$ decays exponentially fast in $n$. We also prove the mean-field lower bound $\mathbb{P}_{\lambda,p}(0\leftrightarrow \infty)\ge c(p-p_c)$ for $p>p_c$.