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:=n1/3|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 n2/3+o(1) vertices.
"Edge percolation on a random regular graph of low degree." Ann. Probab. 36 (4) 1359 - 1389, July 2008. https://doi.org/10.1214/07-AOP361