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

TL;DR

This paper investigates whether the property of a weighted lattice graph having a crossing distance above its median is sensitive to small perturbations, providing the first progress on a question posed by Benjamini in 2011.

Contribution

The paper demonstrates that the property of being above the median in a weighted lattice graph is noise sensitive for a specific crossing distance observable.

Findings

Proves noise sensitivity of the median-exceeding property for a certain crossing distance

Introduces a new observable related to minimal path weights with bounded vertical fluctuation

Provides the first progress on Benjamini's 2011 question about noise sensitivity in this context

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.