The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erdős‐Rényi random graph exceeds its expectation by some constant factor. There has been much exciting recent progress on this problem. We study the corresponding problem for hypergraphs, for which less is known about the large deviation rate. We present new phenomena in upper tail large deviations for sparse random hypergraphs that are not seen in random graphs. We conjecture a formula for the large deviation rate, that is, the first order asymptotics of the log‐probability that the number of copies of fixed subgraph H in a sparse Erdős‐Rényi random k‐uniform hypergraph exceeds its expectation by a constant factor. This conjecture turns out to be significantly more intricate compared to the case for graphs. We verify our conjecture when the fixed subgraph H being counted is a clique, as well as when H is the 3‐uniform 6‐vertex 4‐edge hypergraph consisting of alternating faces of an octahedron, where new techniques are required.

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关于随机超图的上尾问题

随机图中的上尾问题要求估计Erdős-Rényi随机图中某些固定子图的副本数超出其预期某个恒定因子的概率。最近在这个问题上取得了令人兴奋的进展。我们研究超图的相应问题，但对于大偏差率知之甚少。对于稀疏随机超图，我们在上尾大偏差中呈现了新现象，而在随机图中则看不到。我们推测一个大偏差率的公式，即稀疏的Erdős-Rényi随机k中固定子图H的副本数的对数概率的一阶渐近性-统一的超图超出其预期的常数。与图形的情况相比，这种推测变得更加复杂。当计数的固定子图H是一个集团时，以及当H是由八面体的交替面组成的3均匀6顶点4边超图时，我们需要验证新的技术。