Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Daniel Ahlberg and Jeffrey E. Steif

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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.


Soit $f:\{0,1\}^{n}\to\{0,1\}$ une fonction booléenne monotone, et $\{\eta_{p}:p\in[0,1]\}$ le couplage monotone canonique d’éléments de $\{0,1\}^{n}$ choisis selon la mesure produit d’intensité $p\in[0,1]$. Le point aléatoire $p\in[0,1]$ en lequel $f(\eta_{p})$ bascule de $0$ à $1$ est souvent concentré près d’une valeur particulière, présentant ainsi un effet de seuil. Pour une suite de telles fonctions booléennes, nous étudions de plus près la fenêtre de seuil correspondante en considérant la loi limite lorsque $n$ tend vers l’infini (proprement normalisée pour être non-dégénérée) de ce point aléatoire critique où la fonction booléenne bascule. Nous déterminons cette loi pour de nombreuses fonctions booléennes classiques, en portant une attention particulière aux cas de la majorité itérée et des croisements de percolation. Il se trouve que ces lois limites ont des comportements d’une grande variété : en fait, nous montrons que toute mesure de probabilité non-dégénérée sur $\mathbb{R}$ peut être obtenue de cette façon à partir d’une suite bien choisie de fonctions booléennes.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 2135-2161.

Received: 24 August 2015
Revised: 11 August 2016
Accepted: 12 August 2016
First available in Project Euclid: 27 November 2017

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Zentralblatt MATH identifier

Primary: 06E30: Boolean functions [See also 94C10] 60F20: Zero-one laws 60F99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Boolean functions Sharp thresholds Influences Iterated majority function Near-critical percolation


Ahlberg, Daniel; Steif, Jeffrey E. Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 2135--2161. doi:10.1214/16-AIHP786.

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