On the probability of nonexistence in binomial subsets

Frank Mousset, Andreas Noever, Konstantinos Panagiotou, Wojciech Samotij

Research output: Contribution to journalArticlepeer-review

Abstract

Given a hypergraph Γ = (Ω, X) and a sequence p = (pω)ω∈Ω of values in (0, 1), let Ωp be the random subset of Ω obtained by keeping every vertex ω independently with probability pω. We investigate the general question of deriving fine (asymptotic) estimates for the probability that Ωp is an independent set in Γ, which is an omnipresent problem in probabilistic combinatorics. Our main result provides a sequence of upper and lower bounds on this probability, each of which can be evaluated explicitly in terms of the joint cumulants of small sets of edge indicator random variables. Under certain natural conditions, these upper and lower bounds coincide asymptotically, thus giving the precise asymptotics of the probability in question. We demonstrate the applicability of our results with two concrete examples: subgraph containment in random (hyper)graphs and arithmetic progressions in random subsets of the integers.

Original languageEnglish
Pages (from-to)493-525
Number of pages33
JournalAnnals of Probability
Volume48
Issue number1
DOIs
StatePublished - 2020

Keywords

  • Harris's inequality
  • Janson's inequality
  • Joint cumulants

Fingerprint

Dive into the research topics of 'On the probability of nonexistence in binomial subsets'. Together they form a unique fingerprint.

Cite this