Upper tails via high moments and entropic stability

By Matan Harel, Frank Mousset, Wojciech Samotij. Duke mathematical journal, 2019

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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 $G_{n,p}$. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph $H$ in $G_{n,p}$. 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 $G_{n,p}$ for all $p=p(n)$ satisfying $n^{-1}\log n\ll p \ll 1$.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e9df7029de0817b688aea5581588c2c35eebf17f