[Paper Review] Global hypercontractivity and its applications

Peter Keevash, Noam Lifshitz+2 more|arXiv (Cornell University)|Mar 8, 2021

Limits and Structures in Graph Theory73 references20 citations

TL;DR

This paper establishes a global hypercontractivity inequality for $p$-biased measures on the discrete cube, enabling sharp threshold results and $p$-biased analogs of classical theorems in analysis of Boolean functions. The key contribution is a quantitative, sparse-regime version of the KKL theorem and a $p$-biased invariance principle, which yield asymptotically sharp Turán numbers for expanded hypergraphs and resolve conjectures in extremal combinatorics.

ABSTRACT

The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the cube is equipped with the uniform measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general $p$-biased measures. However, simple examples show that when $p = o(1)$, there is no hypercontractive inequality that is strong enough. In this paper, we establish an effective hypercontractive inequality for general $p$ that applies to `global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, thereby making progress on a conjecture of Kahn and Kalai and by establishing a $p$-biased analog of the invariance principle. Our results have significant applications in Extremal Combinatorics. Here we obtain new results on the Turán number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in the area. In particular, we give general conditions under which the crosscut parameter asymptotically determines the Turán number, answering a question of Mubayi and Verstraëte. We also apply the Junta Method to refine our asymptotic results and obtain several exact results, including proofs of the Huang--Loh--Sudakov conjecture on cross matchings and the Füredi--Jiang--Seiver conjecture on path expansions.

Motivation & Objective

• To overcome the failure of classical hypercontractivity in the $p$-biased setting for small $p$, particularly in the sparse regime.
• To develop a hypercontractive inequality that applies to 'global functions'—functions insensitive to small coordinate restrictions—enabling strong structural results.
• To extend the KKL theorem and the invariance principle to the $p$-biased setting with tight quantitative bounds.
• To apply the new tools to obtain asymptotically sharp Turán numbers for bounded-degree uniform hypergraphs expanded from low-uniformity hypergraphs.
• To resolve open problems in extremal combinatorics, including the Huang–Loh–Sudakov and Füredi–Jiang–Seiver conjectures.

Proposed method

• Introduce a new global hypercontractivity inequality tailored for functions that are not significantly affected by fixing a small set of coordinates.
• Use the Junta Method in conjunction with the new hypercontractive inequality to analyze sparse functions and derive sharp threshold behavior.
• Establish a $p$-biased generalization of the Mossel–O’Donnell–Oleszkiewicz invariance principle for low-degree polynomials under $p$-biased measures.
• Apply the theory to hypergraph Turán problems by analyzing the crosscut parameter and generalized criticality of expanded hypergraphs.
• Use contradiction arguments involving cross-free families and matching structures to bound measure differences and derive extremal results.
• Leverage the structure of expanded hypergraphs to reduce Turán number estimation to analyzing base hypergraphs and their expansion parameters.

Experimental results

Research questions

• RQ1 Can a hypercontractive inequality be developed for $p$-biased measures that is effective in the sparse regime ($p \to 0$)?
• RQ2 Can the KKL theorem be strengthened to be both quantitatively tight and applicable when $\mu_p(f) = o(1)$?
• RQ3 Is there a $p$-biased analog of the invariance principle of Mossel, O’Donnell, and Oleszkiewicz?
• RQ4 Can the Turán number of an expanded hypergraph be asymptotically determined by its crosscut parameter?
• RQ5 What conditions ensure that the generalized criticality parameter determines the Turán number for expanded hypergraphs?

Key findings

• The paper establishes a global hypercontractivity inequality that is effective in the sparse $p$-biased regime for global functions.
• It proves a sharp threshold result for monotone global functions, bounding the ratio between the critical probability and the threshold for non-negligible measure within a constant factor.
• A $p$-biased invariance principle is obtained, extending the seminal result of Mossel, O’Donnell, and Oleszkiewicz to general $p$.
• The Turán number of any bounded-degree uniform hypergraph expanded from a bounded-uniformity hypergraph is asymptotically determined by its crosscut parameter over an essentially optimal regime.
• The Huang–Loh–Sudakov conjecture on cross matchings and the Füredi–Jiang–Seiver conjecture on path expansions are proven exactly via the Junta Method and refined asymptotic analysis.
• The results resolve long-standing open problems in extremal combinatorics, including determining the Turán number for expanded hypergraphs under broad conditions.