PreprintCC BY 4.0arXiv
Regularity lemma for Gaussian polynomials
Abstract
We prove a regularity lemma for polynomials with fixed degree d and any number of standard Gaussian input variables. Following modern approaches to the Szemer{é}di regularity lemma, we deduce our results from a general formulation of the regularity lemma using ultraproducts. Along the way, we give a direct combinatorial proof of the celebrated ``fourth moment theorem'' of Nualart and Peccati, which characterizes the asymptotically Gaussian polynomials inside degree-d Gaussian chaos as those such that 𝔼 p2 → 1 and 𝔼 p4 → 3.
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BibTeX
@misc{jones2026regularity,
title = {{Regularity lemma for Gaussian polynomials}},
author = {Chris Jones},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16180},
eprint = {2610.16180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16180},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regularity lemma for Gaussian polynomials
AU - Chris Jones
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16180
UR - https://arxiv.org/abs/2610.16180
AB - We prove a regularity lemma for polynomials with fixed degree d and any number of standard Gaussian input variables. Following modern approaches to the Szemer{é}di regularity lemma, we deduce our results from a general formulation of the regularity lemma using ultraproducts. Along the way, we give a direct combinatorial proof of the celebrated ``fourth moment theorem'' of Nualart and Peccati, which characterizes the asymptotically Gaussian polynomials inside degree-d Gaussian chaos as those such that 𝔼 p^2 → 1 and 𝔼 p^4 → 3.
ER -