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July 2008 Edge percolation on a random regular graph of low degree
Boris Pittel
Ann. Probab. 36(4): 1359-1389 (July 2008). DOI: 10.1214/07-AOP361


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.


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Boris Pittel. "Edge percolation on a random regular graph of low degree." Ann. Probab. 36 (4) 1359 - 1389, July 2008.


Published: July 2008
First available in Project Euclid: 29 July 2008

zbMATH: 1160.05054
MathSciNet: MR2435852
Digital Object Identifier: 10.1214/07-AOP361

Primary: 05432 , 60G42 , 60K35 , 82B27 , 82C20

Keywords: Giant component , percolation , random graph , threshold probability , transition window

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 4 • July 2008
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