Topology of Percolation Clusters: Central Limit Theorems beyond the Lattice
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| Format: | Preprint |
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2026
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| _version_ | 1866908948309737472 |
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| author | de Araújo, Luciano H. L. Machado, Daniel Miranda Coletti, Cristian F. |
| author_facet | de Araújo, Luciano H. L. Machado, Daniel Miranda Coletti, Cristian F. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_07579 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topology of Percolation Clusters: Central Limit Theorems beyond the Lattice de Araújo, Luciano H. L. Machado, Daniel Miranda Coletti, Cristian F. Probability Algebraic Topology 60K35 (Primary) 60F05, 05C80, 20F65, 60D05, 55N35 (Secondary) 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. |
| title | Topology of Percolation Clusters: Central Limit Theorems beyond the Lattice |
| topic | Probability Algebraic Topology 60K35 (Primary) 60F05, 05C80, 20F65, 60D05, 55N35 (Secondary) |
| url | https://arxiv.org/abs/2604.07579 |