Edge percolation on a random regular graph of low degree

TL;DR

This paper analyzes the phase transition in edge percolation on random regular graphs, revealing the precise width of the transition window and detailed component size behavior around the critical probability.

Contribution

It provides a detailed characterization of the transition window and component sizes in percolation on random regular graphs near the critical threshold.

Findings

Transition window width is roughly of order n^{-1/3}.

Largest component size scales as n(p-p^*) when above threshold.

Below threshold, largest component is of size O((p-p^*)^{-2} log n).

Abstract

Consider a uniformly random regular graph of a fixed degree $d\ge3$, with $n$ vertices. Suppose that each edge is open (closed), with probability $p(q=1-p)$, respectively. In 2004 Alon, Benjamini and Stacey proved that $p^*=(d-1)^{-1}$ is the threshold probability for emergence of a giant component in the subgraph formed by the open edges. In this paper we show that the transition window around $p^*$ has width roughly of order $n^{-1/3}$. More precisely, suppose that $p=p(n)$ is such that $\omega:=n^{1/3}|p-p^*|\to\infty$. If $p<p^*$, then with high probability (whp) the largest component has $O((p-p^*)^{-2}\log n)$ vertices. If $p>p^*$, and $\log\omega\gg\log\log n$, then whp the largest component has about $n(1-(p\pi+q)^d)\asymp n(p-p^*)$ vertices, and the second largest component is of size $(p-p^*)^{-2}(\log n)^{1+o(1)}$, at most, where $\pi=(p\pi+q)^{d-1},\pi\in(0,1)$. If $\omega$ is merely polylogarithmic in $n$, then whp the largest component contains $n^{2/3+o(1)}$ vertices.