## 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 H that guarantee that the logarithmic lower tail probabilities for the number of edges, induced by a binomial random subset of the vertices of 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.

Original language | English |
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Pages (from-to) | 665-698 |

Number of pages | 34 |

Journal | Annals of Probability |

Volume | 51 |

Issue number | 2 |

DOIs | |

State | Published - 2023 |

## Keywords

- Lower tail
- random graph
- subgraph counts