Lower tails via relative entropy

TL;DR

This paper demonstrates that the mean-field approximation accurately predicts the leading term of lower tail probabilities for subgraph counts in random graphs and arithmetic progressions in random sets, supported by a new hypergraph degree condition.

Contribution

It introduces sufficient conditions on hypergraph degrees ensuring the mean-field approximation's validity for lower tail probabilities, extending probabilistic methods beyond independent sets.

Findings

Mean-field approximation correctly predicts lower tail probabilities.

New hypergraph degree conditions guarantee approximation accuracy.

Applicable to sparse, dependent structures in random combinatorial objects.

Abstract

We show that the naive mean-field approximation correctly predicts the leading term of the logarithmic lower tail probabilities for the number of copies of a given subgraph in $G(n,p)$ and of arithmetic progressions of a given length in random subsets of the integers in the entire range of densities where the mean-field approximation is viable. Our main technical result provides sufficient conditions on the maximum degrees of a uniform hypergraph $\mathcal{H}$ that guarantee that the logarithmic lower tail probabilities for the number of edges induced by a binomial random subset of the vertices of $\mathcal{H}$ can be well-approximated by considering only product distributions. This may be interpreted as a weak, probabilistic version of the hypergraph container lemma that is applicable to all sparser-than-average (and not only independent) sets.