Classifying the Concentration of the Boolean Cube for Dependent Distributions

TL;DR

This paper investigates how the concentration of the Hamming distance in the Boolean cube varies for dependent distributions, revealing dependence on the choice of point and conditional variances, with a simple, comprehensive proof method.

Contribution

It characterizes the concentration behavior of dependent distributions on the Boolean cube using an inductive bound and correlation conditions, extending understanding beyond product measures.

Findings

Concentration rate depends on the choice of point in the Boolean cube.

Variance of conditional distributions influences concentration quality.

A simple, comprehensive proof technique applies to all Lipschitz functions.

Abstract

A metric probability space $(\Omega,d)$ obeys the ${\it concentration\; of\; measure\; phenomenon}$ if subsets of measure $1/2$ enlarge to subsets of measure close to 1 as a transition parameter $\epsilon$ approaches a limit. In this paper we consider the concentration of the space itself, namely the concentration of the metric $d(x,y)$ for a fixed $y\in \Omega$. For any $y\in \Omega$, the concentration of $d(x,y)$ is guaranteed for product distributions in high dimensions $n$, as $d(x,y)$ is a Lipschitz function in $x$. In fact, in the product setting, the rate at which the metric concentrates is of the same order in $n$ for any fixed $y\in \Omega$. The same thing, however, cannot be said for certain dependent (non-product) distributions. For the Boolean cube $I_n$ (a widely analyzed simple model), we show that, for any dependent distribution, the rate of concentration of the Hamming distance $d_H(x,y)$, for a fixed $y$, depends on the choice of $y\in I_n$, and on the variance of the conditional distributions $\mu(x_k \mid x_1,\dots, x_{k-1})$, $2\leq k\leq n$. We give an inductive bound which holds for all probability distributions on the Boolean cube, and characterize the quality of concentration by a certain positive (negative) correlation condition. Our method of proof is advantageous in that it is both simple and comprehensive. We consider uniform bounding techniques when the variance of the conditional distributions is negligible, and show how this basic technique applies to the concentration of the entire class of Lipschitz functions on the Boolean cube.