Hypercontractivity on HDX II: Symmetrization and q-Norms

2024-08-29

arXiv.org

AI Summary

This work resolves a central open problem: extending Bourgain’s symmetrization theorem to high-dimensional expanders (HDXs). Addressing the lack of suitable tools for hypercontractivity analysis on HDXs, we introduce *q-norm HDX theory*—unifying the characterization of L^q behavior—and *coordinate-component decomposition*, which locally reduces high-dimensional random walks to one-dimensional operators, enabling tractable spectral analysis. We establish the first boosting theorem for HDXs and prove near-optimal (2→q)-hypercontractivity for a broad class of HDXs. These results fully resolve the open question posed by Gur–Lifshitz–Liu (STOC’22), constructing fully hypercontractive subsets with support size n·exp(poly(d)), an exponential improvement over BHKL’22. Our approach integrates Fourier analysis, noise operators, HDX spectral theory, and randomized symmetrization techniques.

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Abstract

Bourgain's symmetrization theorem is a powerful technique reducing boolean analysis on product spaces to the cube. It states that for any product $Omega_i^{otimes d}$, function $f: Omega_i^{otimes d} o mathbb{R}$, and $q>1$: $$||T_{frac{1}{2}}f(x)||_q leq || ilde{f}(r,x)||_{q} leq ||T_{c_q}f(x)||_q$$ where $T_{ ho}f = sumlimits ho^Sf^{=S}$ is the noise operator and $widetilde{f}(r,x) = sumlimits r_Sf^{=S}(x)$ `symmetrizes' $f$ by convolving its Fourier components ${f^{=S}}_{S subseteq [d]}$ with a random boolean string $r in {pm 1}^d$. In this work, we extend the symmetrization theorem to high dimensional expanders (HDX). Building on (O'Donnell and Zhao 2021), we show this implies nearly-sharp $(2{ o}q)$-hypercontractivity for partite HDX. This resolves the main open question of (Gur, Lifshitz, and Liu STOC 2022) and gives the first fully hypercontractive subsets $X subset [n]^d$ of support $ncdotexp( ext{poly}(d))$, an exponential improvement over Bafna, Hopkins, Kaufman, and Lovett's $ncdotexp(exp(d))$ bound (BHKL STOC 2022). Adapting (Bourgain JAMS 1999), we also give the first booster theorem for HDX, resolving a main open question of BHKL. Our proof is based on two elementary new ideas in the theory of high dimensional expansion. First we introduce `$q$-norm HDX', generalizing standard spectral notions to higher moments, and observe every spectral HDX is a $q$-norm HDX. Second, we introduce a simple method of coordinate-wise analysis on HDX which breaks high dimensional random walks into coordinate-wise components and allows each component to be analyzed as a $ extit{$1$-dimensional}$ operator locally within $X$. This allows for application of standard tricks such as the replacement method, greatly simplifying prior analytic techniques.

Problem

• Extends symmetrization theorem to HDX
• Demonstrates nearly-sharp hypercontractivity for HDX
• Introduces q-norm HDX and coordinate-wise analysis

Innovation

• Extends symmetrization theorem to HDX
• Introduces q-norm HDX concept
• Develops coordinate-wise HDX analysis