Upper tail large deviations of regular subgraph counts in Erdős-Rényi graphs in the full localized regime

Abstract

For a ∆-regular graph H the problem of determining the upper tail large deviation for the number of copies of H in G(n, p), an Erdős-Rényi graph on n vertices with edge probability p, has generated significant interests. For p = o(1) and np ∆/2, where v H is the number of vertices in H, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation of the event that the number of copies of H in G(n, p) exceeds its expectation by a constant factor is predicted to hold at a speed n 2 p ∆ log(1/p) and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years covering progressively broader ranges of p, when H is a clique of a fixed size this conjecture was recently settled by Harel, Mousset, and Samotij [12] for the entire localized regime. In this paper we resolve the same for cycles of any given length.

Large deviations in random Latin squares

“…What is the probability it has fewer than half as many as expected? These types of questions have been intensely studied and are intimately related to the development of many important techniques in graph theory and probability theory; see for example the monograph of Chatterjee [15] and the more recent works [1,4,5,16,28]. Beyond Theorem 1.1, we are able to prove the following near-optimal bounds on large deviation probabilities for intercalates in random Latin squares.…”

Sparse random graphs with many triangles

“…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.…”

Lower tails via relative entropy

“…A very different, combinatorial technique for computing upper tail probabilities of polynomials of independent Bernoulli random variables was recently developed by Harel,Mousset,and Samotij [20]. This technique was used to resolve the upper tail problem completely for cliques [20] and, subsequently, for all regular graphs [5]. More precisely, these works showed that the approximation (1) is valid in the entire range of densities p where it was expected to hold.…”

“…otherwise, m 2 (H) := 1 2 . The notation F ⊆ H here means that F is a subgraph of H. For example, m 2 ( ) = 2 but m 2 ( ) = 5 2 because the maximum is attained at the subgraph . Theorem 2.…”