Product Space Models of Correlation: Between Noise Stability and Additive Combinatorics

TL;DR

This paper explores the connection between noise stability and additive combinatorics, establishing bounds and characterizing obstructions for correlation inequalities in product spaces with bounded correlation.

Contribution

It generalizes previous results by addressing the correlation question for multiple functions and bounded correlation, combining analytic and combinatorial methods.

Findings

Established correlation bounds for =2 and >2 with bounded correlation () < 1.

Characterized obstructions to lower bounds in multi-function settings.

Integrated techniques from influences, hyper-contraction, and additive combinatorics.

Abstract

There is a common theme to some research questions in additive combinatorics and noise stability. Both study the following basic question: Let $\mathcal{P}$ be a probability distribution over a space $\Omega^\ell$ with all $\ell$ marginals equal. Let $\underline{X}^{(1)}, \ldots, \underline{X}^{(\ell)}$ where $\underline{X}^{(j)} = (X_1^{(j)}, \ldots, X_n^{(j)})$ be random vectors such that for every coordinate $i \in [n]$ the tuples $(X_i^{(1)}, \ldots, X_i^{(\ell)})$ are i.i.d. according to $\mathcal{P}$. A central question that is addressed in both areas is: - Does there exist a function $c_{\mathcal{P}}()$ independent of $n$ such that for every $f: \Omega^n \to [0, 1]$ with $\mathrm{E}[f(X^{(1)})] = \mu > 0$: \begin{align*} \mathrm{E} \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] \ge c(\mu) > 0 \, ? \end{align*} Instances of this question include the finite field model version of Roth's and Szemer\'edi's theorems as well as Borell's result about the optimality of noise stability of half-spaces. Our goal in this paper is to interpolate between the noise stability theory and the finite field additive combinatorics theory and address the question above in further generality than considered before. In particular, we settle the question for $\ell = 2$ and when $\ell > 2$ and $\mathcal{P}$ has bounded correlation $\rho(\mathcal{P}) < 1$. Under the same conditions we also characterize the _obstructions_ for similar lower bounds in the case of $\ell$ different functions. Part of the novelty in our proof is the combination of analytic arguments from the theories of influences and hyper-contraction with arguments from additive combinatorics.