Vanishing uniqueness thresholds in Voronoi percolation on products

We study Poisson–Voronoi percolation and its discrete analogue Bernoulli–Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold $p_u(\lambda )$ at small intensities $\lambda >0$ based on the unbounded borders phenomenon of their underlining ideal Poisson–Voronoi tessellation. We apply our method to several concrete examples in both the discrete and the continuum setting, including $k$-fold graph products of $d$-regular trees for $k\ge 2,d\ge 3$ and products of hyperbolic spaces ${\mathbb{H}}_{d_1}\times \dots \times {\mathbb{H}}_{d_k}$ for $k\ge 2,d_i\ge 2$, complementing a recent result of the second and fourth author for symmetric spaces of connected higher rank semisimple real Lie groups with property (T). We also provide new examples of non-amenable Cayley graphs with the FIID sparse unique infinite cluster property, answering positively a recent question of Pete and Rokob.