The phase transition in random graphs: A simple proof

Michael Krivelevich, Benny Sudakov

Research output: Contribution to journalArticlepeer-review


The classical result of Erdos and Rényi asserts that the random graph G(n,p) experiences sharp phase transition around \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*}p=\frac{1}{n}\end{align*} \end{document} - for any ε > 0 and \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*}p=\frac{1-\epsilon}{n}\end{align*} \end{document}, all connected components of G(n,p) are typically of size Oε(log n), while for \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*}p=\frac{1+\epsilon}{n}\end{align*} \end{document}, 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 \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*}p=\frac{1+\epsilon}{n}\end{align*} \end{document}, 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.

Original languageEnglish
Pages (from-to)131-138
Number of pages8
JournalRandom Structures and Algorithms
Issue number2
StatePublished - Sep 2013


  • Giant component
  • Long paths
  • Phase transition
  • Random graphs

All Science Journal Classification (ASJC) codes

  • Software
  • Applied Mathematics
  • General Mathematics
  • Computer Graphics and Computer-Aided Design


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