Replica Symmetry in Upper Tails of Mean-Field Hypergraphs

Given a sequence of $s$-uniform hypergraphs $\{H_n\}_{n \geq 1}$, denote by $T_p(H_n)$ the number of edges in the random induced hypergraph obtained by including every vertex in $H_n$ independently with probability $p \in (0, 1)$. 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 $T_p(H_n)$ are precisely approximated by the so-called 'mean-field' variational problem, under certain assumptions on the sequence $\{H_n\}_{n \geq 1}$. In this paper, we study properties of this variational problem for the upper tail of $T_p(H_n)$, 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.