Strong noise sensitivity and random graphs

TL;DR

This paper investigates the noise sensitivity of Boolean functions, especially in the context of Erdős-Rényi random graphs, revealing new insights beyond classical Fourier-based criteria.

Contribution

It introduces a stronger form of noise sensitivity analysis tailored for random graph properties, extending beyond the classical Fourier coefficient approach.

Findings

Noise sensitivity can occur even when the BKS criterion does not apply.

The study applies to graph properties with critical probability tending to zero.

New methods are developed for analyzing noise effects given specific witnesses.

Abstract

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 [Inst. Hautes \'{E}tudes Sci. Publ. Math. 90 (1999) 5-43], 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 $p_c$. 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 Erd\H{o}s-R\'{e}nyi 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, for example, when $p_c\to0$ polynomially fast in the number of variables.