@inproceedings{borgs2004random,
author = {Borgs, Christian and Chayes, Jennifer and Hofstad, Remco van der and Slade, Gordon and Spencer, Joel},
title = {Random Subgraphs of Finite Graphs: III. The Phase Transition for the n-cube},
booktitle = {Combinatorica},
year = {2004},
month = {September},
abstract = {We study random subgraphs of the n-cube f0; 1gn, where nearest-neighbor edges are occupied with probability p. Let pc(n) be the value of p for which the expected size of the component containing a fixed vertex attains the value ¸2n=3, where ¸ is a small positive constant. Let ² = n(p¡pc(n)). In two previous papers, we showed that the largest component inside a scaling window given by j²j = £(2¡n=3) is of size £(22n=3), below this scaling window it is at most 2(log 2)n²¡2, and above this scaling window it is at most O(²2n). In this paper, we prove that for p ¡ pc(n) ¸ e¡cn1=3 the size of the largest component is at least £(²2n), which is of the same order as the upper bound. The proof is based on a method that has come to be known as “sprinkling,” and relies heavily on the specific geometry of the n-cube.},
url = {http://approjects.co.za/?big=en-us/research/publication/random-subgraphs-finite-graphs-iii-phase-transition-n-cube/},
pages = {395-410},
volume = {26},
edition = {Combinatorica},
}