The phase transition in random graphs: A simple proof


  • Research supported in part by a USA-Israel BSF grant and by a grant from the Israel Science Foundation.

  • Research supported in part by NSF grant DMS-1101185, by AFOSR MURI grant FA9550-10-1-0569 and by a USA-Israel BSF grant.


The classical result of Erdős and Rényi asserts that the random graph G(n,p) experiences sharp phase transition around equation image – for any ε > 0 and equation image, all connected components of G(n,p) are typically of size Oε(log n), while for equation image, 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 equation image, 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. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013