A Note on the Probability of Rectangles for Correlated Binary Strings

TL;DR

This paper investigates the probability of correlated binary strings falling into specific sets, showing that Hamming balls approximately maximize or minimize this probability depending on the correlation strength, extending previous results to a new regime.

Contribution

It extends hypercontractive inequality techniques to analyze the probability of correlated binary strings in the regime where set sizes are exponential in n.

Findings

Hamming balls approximately maximize the probability as correlation approaches 1.

Opposite Hamming balls approximately minimize the probability as correlation approaches 0.

Results apply to sets of size exponential in n, extending prior work on large sets.

Abstract

Consider two sequences of $n$ independent and identically distributed fair coin tosses, $X=(X_1,\ldots,X_n)$ and $Y=(Y_1,\ldots,Y_n)$, which are $\rho$-correlated for each $j$, i.e. $\mathbb{P}[X_j=Y_j] = {1+\rho\over 2}$. We study the question of how large (small) the probability $\mathbb{P}[X \in A, Y\in B]$ can be among all sets $A,B\subset\{0,1\}^n$ of a given cardinality. For sets $|A|,|B| = \Theta(2^n)$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|A|,|B| = 2^{\Theta(n)}$. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $\mathbb{P}[X \in A, Y\in B]$ in the regime of $\rho \to 1$. We also prove a similar tight lower bound, i.e. show that for $\rho\to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb{P}[X \in A, Y\in B]$.