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arxiv: 2604.07579 · v1 · submitted 2026-04-08 · 🧮 math.PR · math.AT

Topology of Percolation Clusters: Central Limit Theorems beyond the Lattice

Luciano H. L. de Ara\'ujo , Daniel Miranda Machado , Cristian F. Coletti This is my paper

Pith reviewed 2026-05-10 16:57 UTC · model grok-4.3

classification 🧮 math.PR math.AT
keywords percolationcentral limit theoremquasi-transitive graphssubexponential growthBetti numbersamenable groupsmartingale decompositionstabilization
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The pith

The number of open percolation clusters intersecting a growing ball satisfies a central limit theorem on quasi-transitive graphs of subexponential growth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes central limit theorems for topological functionals of Bernoulli bond percolation on infinite graphs that go beyond the standard lattice. It proves that the count of open clusters touching the metric ball of radius r becomes normally distributed after centering and scaling as r grows, for quasi-transitive graphs with subexponential growth. A broader result covers stationary functionals along Følner sequences on amenable Cayley graphs, under stabilization and moment conditions when the group has a left-orderable finite-index subgroup. The approach is applied to obtain limit theorems for Betti numbers in random simplicial complexes built from the percolation. These findings allow analysis of random cluster topology on a wider class of underlying structures than previously available.

Core claim

For quasi-transitive graphs of subexponential growth, the number K_r of open clusters intersecting the metric ball B_r satisfies a CLT as r→∞. For amenable Cayley graphs, a general CLT holds for stationary percolation functionals along Følner sequences under sequential stabilization and finite-moment assumptions, provided the group admits a left-orderable finite-index subgroup. This applies in particular to groups of polynomial growth. As an application, CLTs are obtained for Betti numbers of graph-generated random simplicial complexes, including clique and neighbor complexes.

What carries the argument

Martingale decompositions arising from invariant edge orderings, together with stabilization estimates that control the effect of single-edge perturbations.

Load-bearing premise

The graph is quasi-transitive with subexponential growth (or an amenable Cayley graph admitting a left-orderable finite-index subgroup) and the functional obeys sequential stabilization plus finite-moment conditions.

What would settle it

Numerical sampling of the normalized cluster count K_r on a concrete quasi-transitive graph of polynomial growth that shows the distribution failing to approach a standard normal for large r.

Figures

Figures reproduced from arXiv: 2604.07579 by Cristian F. Coletti, Daniel Miranda Machado, Luciano H. L. de Ara\'ujo.

Figure 1
Figure 1. Figure 1: Realization of a 1-simplex, a 2-simplex and a 3-simplex of cliques highlighted in a graph. Example 2.2. We say that σ is a neighbor simplex of the graph G if σ ⊆ {v} ∪ NG(v), for some v ∈ V . The neighbor simplicial complex of G is then defined as the collection of all such neighbor simplices in G. Example 2.3 (Weighted Vietoris–Rips complexes). Let G = (V, E) be a locally finite, quasi￾transitive graph wi… view at source ↗
Figure 2
Figure 2. Figure 2: Example of Dr(e, 4) event in Z 2 . Now, we need the following lemma. Lemma 4.2. Let f be the growth function of a quasi-transitive graph. The function f has subexponential growth if and only if there exists a function g : N → N satisfying the following conditions: 1. limr→∞ g(r) = ∞; 2. 0 < g(r) < r for all sufficiently large r; 3. limr→∞ f(r) − f(r − g(r)) f(r) = 0. 4. For any fixed k ≥ 0, limr→∞ f(2g(r)+… view at source ↗
Figure 3
Figure 3. Figure 3: Example of Jr(e, 4) event. ∂B4(x1(e)) x1(e) x2(e) e C ′ 4 (x1(e)) C ′ 4 (x2(e)) [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Lexicographic martingale filtration on Z 2 : the preceding edges (blue) represent the history Fj−1, the red edge is the current edge ej , and the gray edges are the succeed￾ing. The dashed diamond is the dG-ball (graph-distance ball) around ej , depicting the stabilization neighborhood. Note that for each r > 0, since F(Ar) depends only on the edges within Ar, it is independent of any edges outside Ar that… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves central limit theorems for topological functionals of Bernoulli bond percolation on infinite graphs. For quasi-transitive graphs of subexponential growth, the number K_r of open clusters intersecting the metric ball B_r satisfies a CLT as r→∞. For amenable Cayley graphs admitting a left-orderable finite-index subgroup, a general CLT holds for stationary percolation functionals along Følner sequences, assuming sequential stabilization and finite-moment conditions; this applies in particular to groups of polynomial growth. The results are applied to obtain CLTs for Betti numbers of graph-generated random simplicial complexes (clique and neighbor complexes). Proofs combine invariant edge orderings, martingale decompositions, and stabilization estimates for single-edge perturbations.

Significance. If the derivations hold, the work meaningfully extends percolation CLTs from Euclidean lattices to quasi-transitive graphs of subexponential growth and amenable Cayley graphs, providing a general framework under explicit hypotheses (sequential stabilization, finite moments, left-orderable finite-index subgroups). The martingale approach with invariant orderings and the application to Betti numbers of random simplicial complexes are notable strengths, offering tools for topological properties of percolation clusters beyond Z^d.

minor comments (3)
  1. [§2.3] §2.3: the definition of sequential stabilization for functionals could be illustrated with a concrete example (e.g., the cluster-counting functional K_r) to clarify how the single-edge perturbation bound is verified.
  2. [Theorem 4.1] Theorem 4.1: the finite-moment assumption (E[|X_e|^p]<∞ for p>2) is stated but the precise value of p used in the martingale variance bound is not cross-referenced to the stabilization estimate in Lemma 3.4.
  3. The paper would benefit from a short remark comparing the left-orderable-subgroup hypothesis to existing results on invariant orderings in amenable groups (e.g., references to works on left-orderable groups in percolation).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of our work. We are pleased that the referee recognizes the significance of extending central limit theorems for percolation functionals beyond Euclidean lattices to quasi-transitive graphs of subexponential growth and amenable Cayley graphs, as well as the applications to Betti numbers of random simplicial complexes. The recommendation for minor revision is noted, but the report does not raise any specific major comments requiring response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a pure mathematical existence proof establishing conditional central limit theorems for percolation functionals under explicitly listed hypotheses (quasi-transitive subexponential growth, amenability plus left-orderable finite-index subgroup, sequential stabilization, and finite-moment bounds). These hypotheses are used to derive martingale decompositions and stabilization estimates directly from the problem setup, with no data fitting, no self-referential parameter estimation, and no load-bearing self-citations that reduce the claimed results to their own inputs by construction. The derivation chain is self-contained against external benchmarks and does not invoke any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters or invented entities are introduced. The work rests on standard background results in percolation theory, martingale theory, and geometric group theory.

axioms (2)
  • standard math Bernoulli bond percolation is well-defined on any locally finite graph and the cluster-counting functional is measurable.
    Implicit in the statement of the CLT for K_r.
  • standard math Martingale decomposition and stabilization estimates apply under the stated growth and amenability conditions.
    Central to the proof sketch in the abstract.

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Reference graph

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