The phase transition in random graphs  a simple proof
Abstract
The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n  for any \epsilon>0 and p=(1\epsilon)/n, all connected components of G(n,p) are typically of size O(log n), while for p=(1+\epsilon)/n, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime p=(1+\epsilon)/n, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games.
 Publication:

arXiv eprints
 Pub Date:
 January 2012
 arXiv:
 arXiv:1201.6529
 Bibcode:
 2012arXiv1201.6529K
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability;
 05C80;
 60C05