Abstract
Suppose that X is a bounded-degree polynomial with nonnegative coefficients on the p-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of X whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the p-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph . We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph H in . To accommodate readers who are interested in learning the basic method, we include a short, self-contained solution to the upper tail problem for the number of triangles in for all satisfying .
Citation
Matan Harel. Frank Mousset. Wojciech Samotij. "Upper tails via high moments and entropic stability." Duke Math. J. 171 (10) 2089 - 2192, 15 July 2022. https://doi.org/10.1215/00127094-2021-0067
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