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2017 On the chemical distance in critical percolation
Michael Damron, Jack Hanson, Philippe Sosoe
Electron. J. Probab. 22: 1-43 (2017). DOI: 10.1214/17-EJP88

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.

Citation

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Michael Damron. Jack Hanson. Philippe Sosoe. "On the chemical distance in critical percolation." Electron. J. Probab. 22 1 - 43, 2017. https://doi.org/10.1214/17-EJP88

Information

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

zbMATH: 06797885
MathSciNet: MR3698744
Digital Object Identifier: 10.1214/17-EJP88

Subjects:
Primary: 60K35, 82B43

JOURNAL ARTICLE
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