NOISE SENSITIVITY OF BOOLEAN FUNCTIONS AND APPLICATIONS TO PERCOLATION

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 arts 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 I. Let to 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 is an element of (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.