Replica symmetry in upper tails of mean-field hypergraphs

Abstract

Given a sequence of s-uniform hypergraphs {Hn}n≥1, denote by Tp(Hn) the number of edges in the random induced hypergraph obtained by including every vertex in Hn independently with probability p∈(0,1). Recent advances in the large deviations of low complexity non-linear functions of independent Bernoulli variables can be used to show that tail probabilities of Tp(Hn) are precisely approximated by the so-called ‘mean-field’ variational problem, under certain assumptions on the sequence {Hn}n≥1. In this paper, we study properties of this variational problem for the upper tail of Tp(Hn), assuming that the mean-field approximation holds. In particular, we show that the variational problem has a universal replica symmetric phase (where it is uniquely minimized by a constant function), for any sequence of regular s-uniform hypergraphs, which depends only on s. We also analyze the associated variational problem for the related problem of estimating subgraph frequencies in a converging sequence of dense graphs. Here, the variational problems themselves have a limit which can be expressed in terms of the limiting graphon, which gives the exact Bahadur slope of the corresponding estimate.