[Paper Review] Hypercontractivity for global functions and sharp thresholds

Peter Keevash, Noam Lifshitz+2 more|arXiv (Cornell University)|Jun 13, 2019

Limits and Structures in Graph Theory4 citations

TL;DR

This paper establishes a hypercontractive inequality for global Boolean functions under general $p$-biased measures, overcoming limitations of classical hypercontractivity in the sparse regime. It enables sharp threshold results and a $p$-biased invariance principle, strengthening Bourgain's theorem and advancing the Kahn-Kalai conjecture on threshold locations.

ABSTRACT

The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the KKL (Kahn-Kalai-Linial) theorem, Friedgut's junta theorem and the invariance principle of Mossel, O'Donnell and Oleszkiewicz. In these results the cube is equipped with the uniform ($1/2$-biased) 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 $p$ is small there is no hypercontractive inequality that is strong enough for such applications. In this paper, we establish an effective hypercontractivity 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. An additional application of our hypercontractivity theorem, is a $p$-biased analog of the seminal invariance principle of Mossel, O'Donnell, and Oleszkiewicz. In a companion paper, we give applications to the solution of two open problems in Extremal Combinatorics.

Motivation & Objective

• To overcome the failure of classical hypercontractivity for small $p$ in the analysis of Boolean functions.
• To develop a hypercontractive inequality applicable to 'global' functions—those insensitive to small coordinate restrictions—under $p$-biased measures.
• To strengthen Bourgain’s sharp threshold theorem with quantitative tightness and applicability in the sparse regime.
• To establish a $p$-biased analog of the Mossel-O'Donnell-Oleszkiewicz invariance principle.
• To provide tools for resolving open problems in extremal combinatorics and threshold phenomena.

Proposed method

• Introduce the concept of 'global functions' as those not significantly affected by fixing a small set of coordinates.
• Define a $p$-biased noise operator $\mathrm{T}_\rho$ acting on functions over $\{0,1\}^n$ under $\mu_p$ measure.
• Establish a hypercontractive inequality for global functions: $\|\mathrm{T}_\rho f\|_r \leq \|f\|_2$ for $\rho$ depending on $p$ and $r$, with $\rho$ bounded away from zero for small $p$.
• Use the noise stability of global functions and their Fourier structure to derive bounds via hypercontractivity.
• Apply the hypercontractivity result to prove sharp threshold theorems and a $p$-biased invariance principle.
• Leverage the inequality to prove that global, almost monotone functions exhibit sharp thresholds with controlled influence.

Experimental results

Research questions

• RQ1 Can a hypercontractive inequality be established for $p$-biased measures when $p$ is small, for functions that are not local?
• RQ2 Can Bourgain’s sharp threshold theorem be strengthened to be quantitatively tight and applicable in the sparse regime?
• RQ3 Is there a $p$-biased analog of the Mossel-O'Donnell-Oleszkiewicz invariance principle for global functions?
• RQ4 Can the Kahn-Kalai conjecture on threshold location be approached using hypercontractivity for global functions?
• RQ5 Can the influence of small sets of coordinates be bounded for global functions under $p$-biased measures?

Key findings

• A hypercontractive inequality is proven for global functions under $p$-biased measures, with $\rho$ bounded away from zero even for small $p$, enabling applications where classical hypercontractivity fails.
• The hypercontractivity result strengthens Bourgain’s sharp threshold theorem, achieving both quantitative tightness and applicability in the sparse regime.
• A $p$-biased invariance principle is established, generalizing the seminal Mossel-O'Donnell-Oleszkiewicz result to non-uniform measures.
• A sharp threshold result is proven for almost monotone global functions: if $f$ is $(\delta,p,q)$-almost monotone and has a coarse threshold in $[p,q]$, then some set $S$ of size $O(1/\delta)$ has influence at least $\delta$ under $p$ or $q$.
• The method enables progress on the Kahn-Kalai conjecture by bounding the ratio between the critical threshold and the expectation threshold within a constant factor.
• The framework opens new avenues for extremal combinatorics, including potential resolution of open problems on thresholds for graphs with bounded degree.