Strong Noise Sensitivity and Random Graphs

Journal article, 2015

The noise sensitivity of a Boolean function describes its likelihood to flip under small perturbations of its input. Introduced in the seminal work of Benjamini, Kalai and Schramm (1999), it was there shown to be governed by the first level of Fourier coefficients in the central case of monotone functions at a constant critical probability. Here we study noise sensitivity and a natural stronger version of it, addressing the effect of noise given a specific witness in the original input. Our main context is the Erdos-Renyi random graph, where already the property of containing a given graph is sufficiently rich to separate these notions. In particular, our analysis implies (strong) noise sensitivity in settings where the BKS criterion involving the first Fourier level does not apply, e.g., when the critical value goes to 0 polynomially fast in the number of variables.