Topology of Percolation Clusters: Central Limit Theorems beyond the Lattice

Abstract

We prove central limit theorems (CLTs) for topological functionals of Bernoulli bond percolation on infinite graphs beyond the Euclidean lattice ${\mathbb{Z}}^d$. For quasi-transitive graphs of subexponential growth, we show that the number $K_r$ of open clusters intersecting the metric ball $B_r$ satisfies a CLT as $r\to \infty$. For amenable Cayley graphs, we prove a general CLT for stationary percolation functionals along Folner sequences under sequential stabilization and a finite-moment assumption, provided the group admits a left-orderable finite-index subgroup. This applies in particular to groups of polynomial growth. As an application, we obtain CLTs for Betti numbers of graph-generated random simplicial complexes, including clique and neighbor complexes. The proofs combine invariant edge orderings, martingale decompositions, and stabilization estimates for single-edge perturbations.