An estimate for the radial chemical distance in $2d$ critical percolation clusters

TL;DR

This paper estimates the radial chemical distance within two-dimensional critical percolation clusters, constructing a path with volume bounds based on three-arm probabilities, and develops shortcut estimates to refine these bounds.

Contribution

It introduces a novel method to estimate radial distances in 2D critical percolation clusters using three-arm points and shortcut techniques, extending previous work to a radial setting.

Findings

Derived an estimate for the chemical distance in 2D critical percolation

Constructed a path using three-arm points with volume bounds

Developed shortcut estimates to improve distance bounds

Abstract

We derive an estimate for the distance, measured in lattice spacings, inside two-dimensional critical percolation clusters from the origin to the boundary of the box of side length $2n$, conditioned on the existence of an open connection. The estimate we obtain is the radial analogue of the one found in the work of Damron, Hanson, and Sosoe. In the present case, however, there is no lowest crossing in the box to compare to, so we construct a path $\gamma$ from the origin to distance $n$ that consists of "three-arm" points, and whose volume can thus be estimated by $O(n^2\pi_3(n))$. Here, $\pi_3(n)$ is the "three-arm probability" that the origin is connected to distance $n$ by three arms, two open and one dual-closed. We then develop estimates for the existence of shortcuts around an edge $e$ in the box, conditional on $\{e\in \gamma\}$, to obtain a bound of the form $O(n^{2-\delta}\pi_3(n))$ for some $\delta>0$.