What is it about?

Let \(d\ge2\) and let \(\Gamma\) be a lattice in \(\mathbb{R}^d\), meaning a discrete subgroup of rank \(d\). Given a prime \(p\), pick one of the \(p^d\) translates of \(p\Gamma\) in \(\Gamma\) uniformly at random, and call it \(B_p\). Do so independently for all primes. An element of \(\Gamma\) is coloured black if it belongs to some \(B_p\) and white otherwise. There are good reasons to study this random colouring.

How many black (resp. white) connected components are there in this random colouring? For the question to make sense, we need to put a graph structure on \(\Gamma\). Denoting by \(\alpha\) the minimal Euclidean distance between any two distinct points of \(\Gamma\), we declare two vertices of \(\Gamma\) to be adjacent if they lie at Euclidean distance \(\alpha\) from each other. We prove that if \(\Gamma\) is nice, then the number of infinite white (resp. black) clusters is almost surely equal to 1 (resp. 0). Our list of nice lattices includes \(\mathbb{Z}^d\), the \(D_d\)-lattice, \(E_8\), and the Leech lattice. The case of \(\mathbb{Z}^d\) was already obtained in this earlier work but the proof presented here is more direct, simple and elementary.

Arxiv:2509.08452