Edge percolation on a random regular graph of low degree

Abstract

Consider a uniformly random regular graph of a fixed degree d≥3, 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 ω:=n^1/3|p−p^*|→∞. If p<p^*, then with high probability (whp) the largest component has O((p−p^*)⁻²log n) vertices. If p>p^*, and log ω≫log log n, then whp the largest component has about n(1−(pπ+q)^d)≍n(p−p^*) vertices, and the second largest component is of size (p−p^*)⁻²(log n)^1+o(1), at most, where π=(pπ+q)^d−1, π∈(0, 1). If ω is merely polylogarithmic in n, then whp the largest component contains n^2/3+o(1) vertices.