Regularity lemma for Gaussian polynomials

Chris Jones

Home/Archive/arXiv

arXiv · 2610.16180

Regularity lemma for Gaussian polynomials

Chris Jones

View original source ↗Read source document ↗Export metadata ↓

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 $\mathbb{E} p^2 \to 1$ and $\mathbb{E} p^4 \to 3$.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Chris Jones. 2026-10-12. Regularity lemma for Gaussian polynomials. https://arxiv.org/abs/2610.16180

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Free probability via entropic optimal transport

Let $μ$ and $ν$ be compactly supported probability measures on $\mathbb{R}$, and let $μ\boxplus ν$ denote their additive free convolution. We show that for all sufficiently large real $z$, \begin{equation} \int_{-\infty}^\infty \log(z - x) \, (μ\boxplus ν)(\mathrm{d}x) = \sup_Π \left\{ \mathbf{E}_Π[\log(z - (X+Y))] - H(Π\mid μ\otimes ν) \right\}, \end{equation} where the supremum is taken over all couplings $Π$ of $μ$ and $ν$. Analogous formulas hold for multiplicative free convolution $μ\boxtimes ν$ and free compression $[μ]_τ$. In this way, integrals of a log-potential against free convolutions can be expressed as entropic optimal transport problems. The corresponding optimal couplings admit explicit formulas, from which the standard $R$- and $S$-transform descriptions of additive and multiplicative free convolution are recovered. In particular, the optimizer $Π_z$ encodes the subordination equations via couplings of random variables. Our approach is based on a large deviation principle on the symmetric group, combined with the quadrature method of Marcus--Spielman--Srivastava.

math.PR↗

arXiv2026

Scaling limit of boundary random walks: A martingale problem approach

We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under appropriate scaling, the process converges to the general Brownian motion in the $J_1$-Skorokhod topology. The main novelty of our approach lies in a result on the asymptotic behavior of the local time of the boundary random walks, allowing us to derive a CLT result for several Brownian-type limit processes on the half-line.

math.PR↗

arXiv2026

Random Polyhedral Cones I: Distributional Results via Gale Duality

Let $U_1,\ldots,U_n$ be independent random vectors uniformly distributed on the unit sphere $\mathbb S^{d-1}\subseteq\mathbb R^d$, where $n\ge d$, and consider the random polyhedral cone \[ \mathcal W_{n,d}:=\mathop{\mathrm{pos}} (U_1,\ldots,U_n) = \{λ_1 U_1+ \ldots + λ_n U_n: λ_1\geq 0, \ldots, λ_n \geq 0\}. \] We establish several distributional results for $\mathcal W_{n,d}$ and the associated spherical polytope $\mathcal W_{n,d}\cap\mathbb S^{d-1}$. Our main contributions include: (i) Let $α_d$ denote the solid angle of $\mathcal W_{d,d}$ and write $m(d,k):=\mathbb E[α_d^k]$ for its $k$-th moment. We prove the symmetry $m(d,k)=m(k,d)$. As an application, we compute $\mathop{\mathrm{Var}}[α_d]=2^{-d}(d+1)^{-1}-4^{-d}$ and derive a closed formula for the third moment. (ii) For $n=d+1,d+2,d+3$ we determine the probability that $\mathcal W_{n,d}\cap\mathbb S^{d-1}$ is a spherical simplex, a spherical analogue of the classical Sylvester problem. In the case $n=d+2$ we also determine the distribution of the number of vertices of $\mathcal W_{d+2,d}\cap\mathbb S^{d-1}$. (iii) Let $f_\ell(\mathcal W_{n,d})$ denote the number of $\ell$-dimensional faces of $\mathcal W_{n,d}$. We prove a distributional limit theorem for $f_\ell(\mathcal W_{n,d})$ in the regime $n=d+k$ and $\ell=d-q$, where $k,q\in\mathbb N$ are fixed and $d\to\infty$. The limit law is a weighted sum of independent chi squared variables, with weights given by explicit eigenvalues of a convolution operator on the sphere. A unifying ingredient is an explicit coupling producing i.i.d. uniform vectors $U_1,\ldots,U_n\in\mathbb S^{d-1}$ together with i.i.d. uniform vectors $V_1,\ldots,V_n\in\mathbb S^{n-d-1}$ whose associated oriented matroids are Gale dual.

math.PR↗