Electronic Journal of Probability

On the chemical distance in critical percolation

Michael Damron, Jack Hanson, and Philippe Sosoe

Full-text: Open access

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.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 75, 43 pp.

Dates
Received: 9 November 2016
Accepted: 3 August 2017
First available in Project Euclid: 14 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1505354464

Digital Object Identifier
doi:10.1214/17-EJP88

Mathematical Reviews number (MathSciNet)
MR3698744

Zentralblatt MATH identifier
06797885

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Keywords
critical percolation chemical distance scaling exponents

Rights
Creative Commons Attribution 4.0 International License.

Citation

Damron, Michael; Hanson, Jack; Sosoe, Philippe. On the chemical distance in critical percolation. Electron. J. Probab. 22 (2017), paper no. 75, 43 pp. doi:10.1214/17-EJP88. https://projecteuclid.org/euclid.ejp/1505354464


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