Electronic Journal of Probability
- Electron. J. Probab.
- Volume 22 (2017), paper no. 75, 43 pp.
On the chemical distance in critical percolation
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.
Electron. J. Probab., Volume 22 (2017), paper no. 75, 43 pp.
Received: 9 November 2016
Accepted: 3 August 2017
First available in Project Euclid: 14 September 2017
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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