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 note, 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 {\it replica symmetric} phase (where it is uniquely minimized by a constant function), for any sequence of {\it 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.

中文翻译：

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平均场超图上尾的复制对称性

给定$s$-uniform超图$\{H_n\}_{n\geq 1}$的序列，用$T_p(H_n)$表示通过包含$H_n中的每个顶点获得的随机诱导超图中的边数$ 以概率 $p \in (0, 1)$ 独立。独立伯努利变量的低复杂度非线性函数的大偏差的最新进展可用于表明 $T_p(H_n)$ 的尾部概率由所谓的“平均场”变分问题精确逼近，在某些情况下对序列 $\{H_n\}_{n \geq 1}$ 的假设。在本笔记中，我们研究了 $T_p(H_n)$ 上尾的这个变分问题的性质，假设平均场近似成立。特别是，我们证明了变分问题有一个通用的 {\it 副本对称} 阶段（它被一个常数函数唯一地最小化），对于任何 {\it regular} $s$-uniform hypergraphs 序列，它只取决于 $s$。我们还分析了在密集图的收敛序列中估计子图频率的相关问题的相关变分问题。在这里，变分问题本身有一个极限，可以用极限图子来表示。