On the upper tail problem for random hypergraphs

The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erd\H{o}s--R\'enyi 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, i.e., the first order asymptotics of the log-probability that the number of copies of fixed subgraph $H$ in a sparse Erd\H{o}s--R\'enyi 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.