1 April 2024 Regularity method and large deviation principles for the Erdős–Rényi hypergraph
Nicholas A. Cook, Amir Dembo, Huy Tuan Pham
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Duke Math. J. 173(5): 873-946 (1 April 2024). DOI: 10.1215/00127094-2023-0029

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 r2, 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.

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Nicholas A. Cook. Amir Dembo. Huy Tuan Pham. "Regularity method and large deviation principles for the Erdős–Rényi hypergraph." Duke Math. J. 173 (5) 873 - 946, 1 April 2024. https://doi.org/10.1215/00127094-2023-0029

Information

Received: 1 February 2022; Revised: 7 April 2023; Published: 1 April 2024
First available in Project Euclid: 29 April 2024

Digital Object Identifier: 10.1215/00127094-2023-0029

Subjects:
Primary: 05C65 , 05C80 , 15A69 , 60F10

Keywords: hypergraph homomorphism , large deviations , Random tensors , sparse counting lemma , tensor decomposition , tensor norms

Rights: Copyright © 2024 Duke University Press

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Vol.173 • No. 5 • 1 April 2024
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