# Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome? Journal article, 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.

near- critical percolation

sharp thresholds

iterated majority function

Boolean functions

influences

## Author

### Daniel Ahlberg

Instituto Nacional de Matematica Pura E Aplicada, Rio de Janeiro

Uppsala University

### Jeffrey Steif

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

### Gabor Pete

Budapest University of Technology and Economics

#### Annales de linstitut Henri Poincare (B) Probability and Statistics

0246-0203 (ISSN)

Vol. 53 4 2135-2161

Mathematics

### DOI

10.1214/16-AIHP786