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Edge Percolation on a Random Regular Graph of Low Degree

Boris Pittel
The Annals of Probability
Vol. 36, No. 4 (Jul., 2008), pp. 1359-1389
Stable URL: http://www.jstor.org/stable/30242893
Page Count: 31
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Edge Percolation on a Random Regular Graph of Low Degree
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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 $n^\sfarc{-1}{3}$. More precisely, suppose that p = p(n) is such that: $\omega := n^\sfrac{1}{3}\midp-p*\mid \rightarro \infty$. If p < p*, then with high probability (whp) the largest component has O((p - p*)⁻² logn) vertices. If p > p*, and logω ≫ loglogn, then whp the largest component has about $n(1-(p\pi + q)^{d}) \asymp n(p - p*)$ vertices, and the second largest component is of size $(p - p*)^{-2}(logn)^{1+o(1)}$, at most, where $\pi = (p\pi + q)^{d-1}$, π ∈ (0, 1). If ω is merely polylogarithmic in n, then whp the largest component contains $n^{\sfrac{2}{3}}+o(1)$ vertices.

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