We prove that the supercritical phase of Voronoi percolation on \(\mathbb {R}^d\) , \(d\ge 3\) , is well behaved in the sense that for every \(p>p_c(d)\) local uniqueness of macroscopic clusters happens with high probability. As a consequence, truncated connection probabilities decay exponentially fast and percolation happens on sufficiently thick 2D slabs. This is the analogue of the celebrated result of Grimmett and Marstrand for Bernoulli percolation and serves as the starting point for renormalization techniques used to study several fine properties of the supercritical phase.
