Regularity method and large deviation principles for the Erd\H{o}s--R\'enyi hypergraph
Abstract
We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails of homomorphism counts in the -uniform Erd\H{o}s--R\'enyi hypergraph for any fixed , generalizing and improving on previous results for the Erd\H{o}s--R\'enyi graph (). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields tail asymptotics for other nonlinear functionals, such as induced homomorphism counts.
Cite
@article{arxiv.2102.09100,
title = {Regularity method and large deviation principles for the Erd\H{o}s--R\'enyi hypergraph},
author = {Nicholas A. Cook and Amir Dembo and Huy Tuan Pham},
journal= {arXiv preprint arXiv:2102.09100},
year = {2023}
}
Comments
Various minor changes based on feedback from referees. Introduction now includes illustrations of technical results for the concrete example of K_4^3 counts, in particular Theorem 1.5 on the sparse counting lemma. To appear in Duke Math. J