Topology of Percolation Clusters: Central Limit Theorems beyond the Lattice

TL;DR

This paper establishes central limit theorems for topological features of percolation clusters on various infinite graphs, extending classical results beyond Euclidean lattices using advanced probabilistic techniques.

Contribution

It introduces new CLTs for percolation functionals on quasi-transitive and amenable Cayley graphs, including applications to Betti numbers of random simplicial complexes.

Findings

CLTs for the number of clusters intersecting large metric balls.

CLTs for Betti numbers of graph-generated random complexes.

Applicable to groups with polynomial growth and certain stabilization conditions.

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.