On the chemical distance in critical percolation

TL;DR

This paper investigates the chemical distance in two-dimensional critical percolation, showing that the shortest open circuit is asymptotically much smaller than the innermost circuit, indicating a smaller exponent for chemical distance.

Contribution

It provides new asymptotic results on the ratio of shortest to innermost circuits and answers a question about the chemical distance exponent in critical percolation.

Findings

Ratio of shortest to innermost circuit size tends to zero as the annulus size increases.

Ratio of shortest crossing length to lowest crossing length tends to zero in probability.

Chemical distance exponent is strictly smaller than that of the lowest path.

Abstract

We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.