Mathematics – Probability
Scientific paper
1998-11-26
Inst.HautesEtudesSci.Publ.Math.90:5-43,1999
Mathematics
Probability
To appear in Inst. Hautes Etudes Sci. Publ. Math
Scientific paper
It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given. Consider, for example, bond percolation on an $n+1$ by $n$ grid. A configuration is a function that assigns to every edge the value 0 or 1. Let $\omega$ be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges $e$ with $\omega(e)=1$. By duality, the probability for having a crossing is 1/2. Fix an $\epsilon\in(0,1)$. For each edge $e$, let $\omega'(e)=\omega(e)$ with probability $1-\epsilon$, and $\omega'(e)=1-\omega(e)$ with probability $\epsilon$, independently of the other edges. Let $p(\tau)$ be the probability for having a crossing in $\omega$, conditioned on $\omega'=\tau$. Then for all $n$ sufficiently large, $P\{\tau : |p(\tau)-1/2|>\epsilon\}<\epsilon$.
Benjamini Itai
Kalai Gil
Schramm Oded
No associations
LandOfFree
Noise Sensitivity of Boolean Functions and Applications to Percolation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Noise Sensitivity of Boolean Functions and Applications to Percolation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Noise Sensitivity of Boolean Functions and Applications to Percolation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-626875