LOWER TAILS VIA RELATIVE ENTROPY

• 成果类型: Article
• 署名作者: Kozma, Gady; Samotij, Wojciech
• 署名单位: Weizmann Institute of Science; Tel Aviv University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1610
• 发表日期: 2023
• 页码: 665-698
• 关键词: Approximation; matrices

摘要

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 ?-L that guarantee that the logarithmic lower tail probabilities for the number of edges, induced by a binomial random sub-set of the vertices of ?-L, can be well approximated by considering only prod-uct 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.