On the upper tail problem for random hypergraphs
Name
1910.02916.pdf
Description
Accepted version
Size
425.15 KB
Format
Adobe PDF
Checksum (MD5)
a001ab9e4f5c6f466216cbb9b6c7f19a
Author(s)
Liu, Yang P
Zhao, Yufei
Date Issued
2021
Journal
Random Structures and Algorithms
Publisher
Wiley
Version
Author's final manuscript
Abstract
© 2020 Wiley Periodicals LLC 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1002/rsa.20975
