Recent work on chemical distance in critical percolation

Abstract

In this note, we describe some of the progress recently made on questions regarding the chemical distance in two-dimensional critical percolation by the author, J. Hanson, and P. Sosoe [6, 7]. It is expected that the distance between points in critical percolation clusters scales as $\|\cdot \|^{1+s}$, where $\|\cdot \|$ is the Euclidean distance and $s>0$. First, we review previous work of Aizenman-Burchard and Morrow-Zhang, which together establish a version of $0 < s \leq 1/3$. The main results of our work are in the direction of proving upper bounds on $s$, answering in [6] a question from '93 of Kesten-Zhang on the ratio of the length of the shortest crossing of a box to the length of the lowest crossing of a box. The paper [7] provides a quantitative version of the result of [6], along with bounds on point-to-point and point-to-set distances.