Remarks on Central Limit Theorems for the Number of Percolation Clusters

Abstract

We consider percolation problems on regular trees and some pre-fractal graphs. Central limit theorems for the number of open clusters in a finite box are obtained. For regular trees and some classes of Sierpiński carpet lattices, we can prove the central limit theorems for all p ∈ (0, 1). §1. Introduction and ResultsCentral limit theorems (CLT's) for percolation problems have been studied by many authors (see Grimmett [5]. Using this method together with the ergodic theorem, Penrose [6] proved a general CLT which can be applied to several models. In this note, by using the argument in [9] we study a CLT for percolation problems on regular trees and Sierpiński carpet lattices, where the ergodic theorem is not available.

Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs and percolation

“…Our next result is a Berry-Esseen bound for the normalized random variable H n (p) := (C n (p)−E[C n (p)])/ Var[C n (p)]. This adds to the qualitative central limit theorem in [37]. Theorem 1.4.…”

“…We use a lower bound for the variance of C n (p), which can be found in [37,Identity (2.3)] in case of a D-regular tree, but the proof is easily seen to carry over to our situation. More precisely, there exists a constant c(p) > 0 only depending on p such that In case of a D-regular tree, we have that |T n | = D + .…”

“…For n ∈ N, we denote by C n (p) the number of connected components of the random graph T n (p), where, as already discussed in the introduction, by a connected component we understand a maximal connected sub-graph with at least one edge. By We use a lower bound for the variance of C n (p), which can be found in [37,Identity (2.3)] in case of a D-regular tree, but the proof is easily seen to carry over to our situation. More precisely, there exists a constant c(p) > 0 only depending on p such that…”