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January 2020 On the probability of nonexistence in binomial subsets
Frank Mousset, Andreas Noever, Konstantinos Panagiotou, Wojciech Samotij
Ann. Probab. 48(1): 493-525 (January 2020). DOI: 10.1214/19-AOP1371

Abstract

Given a hypergraph $\Gamma =(\Omega ,\mathcal{X})$ and a sequence $\mathbf{p}=(p_{\omega })_{\omega \in \Omega }$ of values in $(0,1)$, let $\Omega_{\mathbf{p}}$ be the random subset of $\Omega $ obtained by keeping every vertex $\omega $ independently with probability $p_{\omega }$. We investigate the general question of deriving fine (asymptotic) estimates for the probability that $\Omega_{\mathbf{p}}$ is an independent set in $\Gamma $, 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.

Citation

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Frank Mousset. Andreas Noever. Konstantinos Panagiotou. Wojciech Samotij. "On the probability of nonexistence in binomial subsets." Ann. Probab. 48 (1) 493 - 525, January 2020. https://doi.org/10.1214/19-AOP1371

Information

Received: 1 November 2017; Revised: 1 April 2019; Published: January 2020
First available in Project Euclid: 25 March 2020

zbMATH: 07206766
MathSciNet: MR4079444
Digital Object Identifier: 10.1214/19-AOP1371

Subjects:
Primary: 05C65, 05C69, 05C80, 60C05

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 1 • January 2020
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