Edge percolation on a random regular graph of low degree

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*, 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 e (0, 1). If ω is merely polylogarithmic in n, then whp the largest component contains n 2/3+o(1) vertices.

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