Regularity method and large deviation principles for the Erdős--Rényi 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 $r$-uniform Erdős--Rényi hypergraph for any fixed $r\ge 2$, generalizing and improving on previous results for the Erdős--Rényi graph ($r=2$). 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.

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