Edge percolation on a random regular graph of low degree

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.