On the chemical distance in critical percolation II

TL;DR

This paper advances understanding of the chemical distance in 2D critical percolation clusters by providing new estimates, demonstrating infinite second moments for point-to-point distances, and quantitatively comparing crossing lengths.

Contribution

It introduces new estimates involving three-arm probabilities and extends previous results on crossing lengths in critical percolation.

Findings

Point-to-point distance has infinite second moment.

New estimates involving three-arm probabilities.

Quantitative comparison of shortest and lowest crossing lengths.

Abstract

We continue our study of the chemical (graph) distance inside large critical percolation clusters in dimension two. We prove new estimates, which involve the three-arm probability, for the point-to-surface and point-to-point distances. We show that the point-to-point distance in $\mathbb{Z}^2$ between two points in the same critical percolation cluster has infinite second moment. We also give quantitative versions of our previous results comparing the length of the shortest crossing to that of the lowest crossing of a box.