  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.
