Is 'being above the median' a noise sensitive property?

Abstract

Assign independent weights to the edges of the square lattice, from the uniform distribution on $\{a,b\}$ for some $0<a<b<\infty$. The weighted graph induces a random metric on $\mathbb{Z}^2$. Let $T_n$ denote the distance between $(0,0)$ and $(n,0)$ in this metric. The distribution of $T_n$ has a well-defined median. Itai Benjamini asked in 2011 if the sequence of Boolean functions encoding whether $T_n$ exceeds its median is noise sensitive? In this paper we present the first progress on Benjamini's problem. More precisely, we study the minimal weight along any path crossing an $n\times n$-square horizontally and whose vertical fluctuation is smaller than $n^{1/22}$, and show that for this observable, 'being above the median' is a noise sensitive property.