The Annals of Probability

Edge percolation on a random regular graph of low degree

Boris Pittel

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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|pp*|→∞. If p<p*, then with high probability (whp) the largest component has O((pp*)−2log n) vertices. If p>p*, and log ω≫log log n, then whp the largest component has about n(1−(+q)d)≍n(pp*) vertices, and the second largest component is of size (pp*)−2(log n)1+o(1), at most, where π=(+q)d−1, π∈(0, 1). If ω is merely polylogarithmic in n, then whp the largest component contains n2/3+o(1) vertices.

Article information

Ann. Probab., Volume 36, Number 4 (2008), 1359-1389.

First available in Project Euclid: 29 July 2008

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Primary: 05432 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B27: Critical phenomena 60G42: Martingales with discrete parameter 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Percolation random graph threshold probability transition window giant component


Pittel, Boris. Edge percolation on a random regular graph of low degree. Ann. Probab. 36 (2008), no. 4, 1359--1389. doi:10.1214/07-AOP361.

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