On the variational problem for upper tails in sparse random graphs

Eyal Lubetzky, Yufei Zhao

Research output: Contribution to journalArticlepeer-review


What is the probability that the number of triangles in Gn,p, the Erdős-Rényi random graph with edge density p, is at least twice its mean? Writing it as exp[-r(n,p)], already the order of the rate function r(n, p) was a longstanding open problem when p = o(1), finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that r(n,p) ≍n2p2log (1/p)for p ≳ logn/n; the exact asymptotics of r(n, p) remained unknown. The following variational problem can be related to this large deviation question at p ≳ logn/n; : for δ > 0 fixed, what is the minimum asymptotic p-relative entropy of a weighted graph on n vertices with triangle density at least (1 + δ)p3? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this problem for fixed p. A very recent breakthrough of Chatterjee and Dembo extended its validity to n for an explicit α > 0, and plausibly it holds in all of the above sparse regime. In this note we show that the solution to the variational problem is {1/2 δ2/3, 1/3 δ when n-1/2 ≪ p ≪ 1 vs. 1/2 δ2/3 when n-1 ≪ p ≪ n-1/2(the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that gn,p for n≤ p ≪ 1 has twice as many triangles as its expectation is exp[-r(n,p)] where r(n,p)~ 1/3 n2p2 log(1/p) Our results further extend to k-cliques for any fixed k, as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when p ≥n.

Original languageEnglish (US)
Pages (from-to)420-436
Number of pages17
JournalRandom Structures and Algorithms
Issue number3
StatePublished - May 1 2017


  • large deviations
  • sparse random graphs
  • upper tails of subgraph counts

ASJC Scopus subject areas

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


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