Components, large and small, are as they should be II: supercritical percolation on regular graphs of constant degree

TL;DR

This paper analyzes supercritical percolation on regular graphs, showing that the largest component size concentrates around a predictable value and smaller components remain relatively small, generalizing previous results.

Contribution

It extends and improves upon existing results by providing precise bounds on component sizes in supercritical percolation on regular graphs under certain conditions.

Findings

Largest component size concentrates around y*n with high probability

Other components are of size O(log n) under specified conditions

Results generalize and improve previous findings for random d-regular graphs

Abstract

Let $d\ge 3$ be a fixed integer. Let $y:= y(p)$ be the probability that the root of an infinite $d$-regular tree belongs to an infinite cluster after $p$-bond-percolation. We show that for every constants $b,\alpha>0$ and $1<\lambda< d-1$, there exist constants $c,C>0$ such that the following holds. Let $G$ be a $d$-regular graph on $n$ vertices, satisfying that for every $U\subseteq V(G)$ with $|U|\le \frac{n}{2}$, $e(U,U^c)\ge b|U|$ and for every $U\subseteq V(G)$ with $|U|\le \log^Cn$, $e(U)\le (1+c)|U|$. Let $p=\frac{\lambda}{d-1}$. Then, with probability tending to one as $n$ tends to infinity, the largest component $L_1$ in the random subgraph $G_p$ of $G$ satisfies $\left|1-\frac{|L_1|}{yn}\right|\le \alpha$, and all the other components in $G_p$ are of order $O\left(\frac{\lambda\log n}{(\lambda-1)^2}\right)$. This generalises (and improves upon) results for random $d$-regular graphs.