Hypercontractivity on HDX II: Symmetrization and q-Norms
Max Hopkins
TL;DR
This paper extends hypercontractivity beyond the Boolean cube to high dimensional expanders (HDX) via Bourgain-like symmetrization. The authors introduce q-norm HDX and a coordinate-wise analysis that decouples high-dimensional walks into tractable 1-D components, enabling a near-sharp $(2\to q)$-hypercontractive inequality for partite HDX and a booster theorem. The key technical contributions are (i) a BV-type symmetrization for HDX that preserves higher-norm structure up to a $(1+o(1))$ factor, and (ii) a global hypercontractivity bound that matches the classical product-space bounds up to poly-log factors, yielding very small support growth, $|X| \approx n \cdot 2^{\text{poly}(d)}$. These results imply strong pseudo-randomness properties for sparse HDX subsets and lead to tighter KKL-type statements and booster phenomena, with potential implications for PCPs and related combinatorial constructions. Overall, the work significantly narrows the gap between hypercontractivity on dense cubes and sparse HDX, providing simpler proofs and sharper parameter regimes that enhance applicability to coding theory, PCPs, and complexity-theoretic constructions.
Abstract
Bourgain's symmetrization theorem is a powerful technique reducing boolean analysis on product spaces to the cube. It states that for any product $Ω_i^{\otimes d}$, function $f: Ω_i^{\otimes d} \to \mathbb{R}$, and $q > 1$: $$||T_{\frac{1}{2}}f(x)||_q \leq ||\tilde{f}(r,x)||_{q} \leq ||T_{c_q}f(x)||_q$$ where $T_ρf = \sum\limits ρ^Sf^{=S}$ is the noise operator and $\widetilde{f}(r,x) = \sum\limits r_Sf^{=S}(x)$ `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 $(2{\to}q)$-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 $n\cdot\exp(\text{poly}(d))$, an exponential improvement over Bafna, Hopkins, Kaufman, and Lovett's $n\cdot\exp(\exp(d))$ 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 $\textit{$1$-dimensional}$ operator locally within $X$. This allows for application of standard tricks such as the replacement method, greatly simplifying prior analytic techniques.