UPPER TAILS VIA HIGH MOMENTS AND ENTROPIC STABILITY

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 D p.n/ satisfying n^-1 log n ≦ p ≦ 1.