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

Abstract

Consider a monotone Boolean function $f:\{0,1\}^n\to\{0,1\}$ and the canonical monotone coupling $\{η_p:p\in[0,1]\}$ 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 $f(η_p)$ 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.

33 pages, 3 figure. A paper by Daniel Ahlberg and Jeffery E. Steif, with an appendix by Gábor Pete. Added to the second version is the appendix (earlier published as arXiv:1507.05522) and a significant strengthening of Theorem 3, obtained jointly with Anders Martinsson. The third version updates the conjecture of the appendix