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 twodimensional 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 AizenmanBurchard and MorrowZhang, 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 KestenZhang 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 pointtopoint and pointtoset distances.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 DOI:
 10.48550/arXiv.1602.00775
 arXiv:
 arXiv:1602.00775
 Bibcode:
 2016arXiv160200775D
 Keywords:

 Mathematics  Probability
 EPrint:
 7 pages