Noise Sensitivity on the p -Biased Hypercube

Abstract

The noise sensitivity of a Boolean function measures how susceptible the value of f on a typical input x to a slight perturbation of the bits of x: it is the probability f(x) and f(y) are different when x is a uniformly chosen n-bit Boolean string, and y is formed by flipping each bit of x with small probability ε. The noise sensitivity of a function is a key concept with applications to combinatorics, complexity theory, learning theory, percolation theory and more. In this paper, we investigate noise sensitivity on the p-biased hypercube, extending the theory for polynomially small p. Specifically, we give sufficient conditions for monotone functions with large groups of symmetries to be noise sensitive (which in some cases are also necessary). As an application, we show that the 2-SAT function is noise sensitive around its critical probability. En route, we study biased versions of the invariance principle for monotone functions and give p-biased versions of Bourgain's tail theorem and the Majority is Stablest theorem, showing that in this case the correct analog of ``small low degree influences'' is lack of correlation with constant width DNF formulas.