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**Harvard**

Ahlberg, D., Steif, J. och Pete, G. (2015) * Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome? *.

** BibTeX **

@unpublished{

Ahlberg2015,

author={Ahlberg, Daniel and Steif, Jeffrey and Pete, Gabor},

title={ 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
{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. },

year={2015},

keywords={Boolean functions; sharp thresholds; influences; iterated majority function; near- critical percolation},

note={30},

}

** RefWorks **

RT Unpublished Material

SR Electronic

ID 229075

A1 Ahlberg, Daniel

A1 Steif, Jeffrey

A1 Pete, Gabor

T1 Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?

YR 2015

AB 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.

LA eng

LK http://www.math.chalmers.se/~steif/p59.pdf

OL 30