## Noise Sensitivity of Boolean Functions And Applications to Percolation (1998)

Citations: | 72 - 16 self |

### BibTeX

@MISC{Benjamini98noisesensitivity,

author = {Itai Benjamini and Gil Kalai and Oded Schramm},

title = {Noise Sensitivity of Boolean Functions And Applications to Percolation},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 ! 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 !(e) = 1. By duality, the probability for having a crossing is 1=2. Fix an ffl 2 (0; 1). For each edge e, let ! 0 (e) = !(e) with probability 1 \Gamma ffl, and ! 0 (e) = 1 \Gamma !(e) with probability ffl, independently of the other edg...