Recent work on chemical distance in critical percolation

TL;DR

This paper reviews recent progress on understanding the chemical distance in two-dimensional critical percolation, including bounds on the scaling exponent and ratios of crossing lengths, advancing the theoretical understanding of percolation geometry.

Contribution

The paper advances the understanding of chemical distance scaling in critical percolation by establishing upper bounds on the exponent and providing quantitative bounds on crossing distances.

Findings

Established that the scaling exponent s satisfies 0 < s ≤ 1/3.

Provided bounds on the ratio of shortest to lowest crossing lengths.

Presented quantitative bounds on point-to-point and point-to-set distances.

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.