Regularity method and large deviation principles for the Erd\H{o}s--R\'enyi hypergraph

TL;DR

This paper establishes a quantitative large deviations framework for Erdős–Rényi hypergraphs using tensor decomposition and novel cut norms, providing sharp asymptotics for tail probabilities of homomorphism counts.

Contribution

It introduces a new large deviations theory for hypergraphs based on tensor methods and cut norms, extending previous graph results to hypergraphs with polynomially vanishing densities.

Findings

Derived sharp asymptotics for tail probabilities of homomorphism counts.

Extended large deviations principles to hypergraphs with polynomially vanishing densities.

Provided tail asymptotics for nonlinear functionals like induced homomorphism counts.

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 Erd\H{o}s--R\'enyi hypergraph for any fixed $r\ge 2$, generalizing and improving on previous results for the Erd\H{o}s--R\'enyi 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.