Upper tails via high moments and entropic stability

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 Gn,p. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph H in Gn,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 Gn,p for all p = p(n) satisfying n −1 log n p 1. Contents 42 8. The Poisson regime 50 9. Beyond polynomials with nonnegative coefficients 58 10. Concluding remarks 59 References 61

Sparse random graphs with many triangles

“…To see where it comes from, note that if, in a graph G, there is a set S ⊆ [n] with |S| = k n such that S is an union of vertex-disjoint triangles, no other edge is present in [S], and V \S is triangle free, then clearly V T (G) ≥ k n , and it is relatively straightforward to obtain a lower bound on P(V T (G n,pn ) ≥ k n ). However, V T (G) does not seem to be a bounded-degree polynomial function of the entries of the adjacency matrix of G. Therefore, unlike T , the theory of non-linear large deviations in Harel, Mousset, and Samotij [21] is not readily applicable in this case. To derive an upper bound we instead develop some new combinatorial tools, in particular, a graph decomposition lemma that we believe to be of independent interest.…”

“…When H = K r is an r-clique (complete subgraph on r vertices), the mis-matching factor log(1/p) was removed by Chatterjee [11] and by DeMarco and Kahn [15], who proved the correct order of magnitude of the logarithmic upper tail probability. Finally, after a series of improvements (Augeri [2], Chatterjee and Dembo [12], Cook and Dembo [14], Eldan [16]), Harel, Mousset, and Samotij [21] obtained the estimate of the form…”

“…Bernoulli random variables. Since then the study of non-linear large deviations have received vast attention (see Augeri [2], Basak and Basu [4], Chatterjee and Dembo [12], Cook and Dembo [14], Eldan [16], Harel, Mousset, and Samotij [21]). Unfortunately, to our knowledge none of these considers the sparse regime p = p n = λ/n.…”

“…The probability that [(6k n ) 1/3 ] forms a clique is roughly exp − 1 2 [6k n ] 2/3 log(1/p n ) . The matching upper bound expressed by (1.2) is more difficult, and to derive it we adopt the theory of non-linear large deviations developed in Harel, Mousset, and Samotij [21]. As mentioned before, the study of small subgraphs is a classical topic in random graph theory.…”

“…Although the expression in (1.11) is much cleaner, the statements in Theorems 2.1 and 3.1 explicitly spell out the error terms. The proofs of these theorems are based on the theory of non-linear large deviations developed in Harel, Mousset, and Samotij [21] (which was partly based on Janson [26]).…”