Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Monday, May 07, 2012, 12:15 pm

Duration: 45 minutes

Location: OAT S15/S16/S17

Speaker: Benny Sudakov (University of California, Los Angeles)

The phase transition in random graphs -- a simple proof

The classical result of Erdős and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n -- for any ε>0 and p=(1-ε)/n, all connected components of G(n,p) are typically of size O(\log n), while for p=(1+ε)/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+ε)/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.

Joint work with M. Krivelelvich.

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