Hypercontractivity on HDX II: Symmetrization and q-Norms (2408.16687v1)

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 $f: \Omega_i^{\otimes d} \to \mathbb{R}$, and $q > 1$: $$||T_{\frac{1}{2}}f(x)||q \leq ||\tilde{f}(r,x)||{q} \leq ||T_{c_q}f(x)||q$$ where $T{\rho}f = \sum\limits \rho^Sf^{=S}$ 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). Adapting work of O'Donnell and Zhao (2021), we give as a corollary a proof of optimal global hypercontractivity for partite HDX, resolving one of the main open questions of Gur, Lifshitz, and Liu (STOC 2022). Adapting work of Bourgain (JAMS 1999), we also give the first booster theorem for HDX, resolving a main open questions of Bafna, Hopkins, Kaufman, and Lovett (STOC 2022). 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 $1$-dimensional operator locally within $X$. This allows for application of standard tricks such as the replacement method, greatly simplifying prior analytic techniques.

• The paper extends classical symmetrization techniques to high-dimensional expanders, proving an optimal (2-to-4)-hypercontractive inequality that resolves a key open question.
• It introduces the concept of q-norm HDX which generalizes spectral notions to higher moments, enabling direct manipulation of higher norms without significant error accumulation.
• The study also extends Bourgain's booster theorem to HDX, revealing that low influence functions on strong HDX complexes exhibit substantial deviations across many restrictions.

Hypercontractivity on HDX II: Symmetrization and q-Norms

The paper "Hypercontractivity on HDX II: Symmetrization and $q$-Norms" by Max Hopkins extends the foundational results in the field of boolean function analysis beyond the cube, specifically targeting high-dimensional expanders (HDX). The primary contributions of this paper center around extending techniques from product spaces to HDX, thereby addressing several key open questions and improving our understanding of hypercontractivity in sparse systems.

At the heart of this research is the generalization of Bourgain's symmetrization theorem to high-dimensional expanders. The classical symmetrization theorem is a pivotal tool in boolean analysis, enabling the reduction of boolean function analysis in product spaces to the (much simpler) cube. This paper successfully extends these insights to HDX, with Hopkins proving the symmetrization theorem for sufficiently strong HDX complexes up to a factor of $(1 \pm o(1))$. This symmetrization theorem states that for any $d$-partite $\gamma$-product satisfying $\gamma \leq 2^{-\Omega_q(d)}$, function $f: X \to R$, and $q > 1$, we have:

$(1-o_{\gamma}(1)){\widetilde{T_{c_q}f}_q \leq {f}_q \leq (1+o_{\gamma}(1)){\widetilde{T_{2}f}_q$

This is a significant theoretical advancement as it provides a powerful tool for translating classical results from product spaces to HDX.

The introduction of `$q$-norm HDX' is another critical contribution of the paper. This concept generalizes standard spectral notions to higher moments, enabling the manipulation of higher norms directly without accumulating problematic error terms that have been a limitation in earlier works. Hopkins shows that any strong enough spectral HDX is also a $q$-norm HDX, an insight that simplifies prior analytic techniques by allowing the use of standard tricks, such as the replacement method.

The practical implications of these theoretical advancements are manifold. First, the symmetrization theorem for HDX leads directly to an optimal global hypercontractive inequality for partite HDX, thus resolving a main open question posed by Gur, Lifshitz, and Liu (STOC 2022). This optimal $(2 \to 4)$-hypercontractive inequality matches the eigenvalues of the noise operator, allowing one to transfer this bound to a variant of the classical operator form of hypercontractivity. Specifically, it shows that for any $d$-partite $\gamma$-product with $\gamma \leq 2^{-\Omega(d)}$, any function $f: X \to R$ satisfies:

${f^{\leq i}_4 \leq 2^{O(i)}{f^{\leq i}_2 \max_{|S| \leq i, x_S} f|_{x_S}_2$

Second, this paper makes strides in understanding the structure of low influence functions on HDX. Hopkins extends Bourgain's classical booster theorem to HDX, demonstrating that any low influence function on a sufficiently strong HDX complex has many restrictions on which it deviates substantially from its expectation. This result is not only theoretically compelling but also opens new avenues for exploring sharp thresholds for graph properties within the framework of high-dimensional expanders.

In the broader context of boolean analysis and theoretical computer science, these results have potential applications in areas such as extremal and probabilistic combinatorics, quantum communication, approximate sampling, and hardness of approximation. The application of symmetrization and hypercontractivity to these fields could yield new insights and algorithms, particularly in improving the efficiency and efficacy of probabilistic and combinatorial bounds.

Future directions for research include further refinement and generalization of these methods to other non-product spaces, such as the Grassmannian, and Lie groups, where hypercontractive inequalities remain prohibitively complex. Another intriguing direction is the exploration of symmetrization techniques in fully non-partite or non-spectral HDX contexts and their implications for combinatorial structures and algorithms.

In conclusion, the paper "Hypercontractivity on HDX II: Symmetrization and $q$-Norms" by Max Hopkins represents a substantial advance in the paper of hypercontractivity in high-dimensional expanders. By extending classical symmetrization techniques to HDX and introducing the notion of $q$-norm HDX, the paper resolves significant open questions and sets the stage for further theoretical and practical developments in boolean function analysis and beyond.