Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?
Artikel i vetenskaplig tidskrift, 2017

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 majorityand percolation crossings. It turns out that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate probability measure on R arises in this way for some sequence of Boolean functions.

iterated majority function

Boolean functions

near- critical percolation

sharp thresholds



Daniel Ahlberg

Instituto Nacional de Matematica Pura E Aplicada, Rio de Janeiro

Uppsala Universitet

Jeffrey Steif

Chalmers, Matematiska vetenskaper, Matematik

Göteborgs universitet

Gabor Pete

Budapest University of Technology and Economics

Alfred Renyi Institute of Mathematics Hungarian Academy of Sciences

Annales de linstitut Henri Poincare (B) Probability and Statistics

0246-0203 (ISSN)

Vol. 53 2135-2161