Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?

TL;DR

This paper investigates the limiting distribution of the threshold point where monotone Boolean functions switch from 0 to 1, revealing diverse behaviors and showing any nondegenerate distribution can be realized through such functions.

Contribution

It characterizes the limiting distributions of threshold windows for various Boolean functions, including iterated majority and percolation, and demonstrates the universality of possible distributions.

Findings

Limiting distributions vary significantly among different Boolean functions.

For some functions, the threshold distribution converges to known probability measures.

Any nondegenerate probability measure on the real line can be obtained as a limit for some sequence of Boolean functions.

Abstract

Consider a monotone Boolean function $f:\{0,1\}^n\to\{0,1\}$ and the canonical monotone coupling $\{\eta_p:p\in[0,1]\}$ of an element in $\{0,1\}^n$ chosen according to product measure with intensity $p\in[0,1]$. The random point $p\in[0,1]$ where $f(\eta_p)$ flips from $0$ to $1$ is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large $n$, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. We determine this distribution for a number of the Boolean functions which are typically studied and pay particular attention to the functions corresponding to iterated majority and percolation crossings. It turns out that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate probability measure on $\mathbb{R}$ arises in this way for some sequence of Boolean functions.