On the upper tail problem for random hypergraphs
Abstract
The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an ErdősRényi 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 logprobability that the number of copies of fixed subgraph $H$ in a sparse ErdősRényi 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 3uniform 6vertex 4edge hypergraph consisting of alternating faces of an octahedron, where new techniques are required.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.02916
 Bibcode:
 2019arXiv191002916L
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability
 EPrint:
 36 pages, v2, to appear in Random Structures &