Is being above the median a Noise Sensitive property?

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.