On the chemical distance in critical percolation

Michael Damron Indiana University Jack Hanson Indiana University Philippe Sosoe Harvard University CMSA

Publications of CMSA of Harvard mathscidoc:1702.38043

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.
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@inproceedings{michaelon,
  title={On the chemical distance in critical percolation},
  author={Michael Damron, Jack Hanson, and Philippe Sosoe},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207032644244050243},
}
Michael Damron, Jack Hanson, and Philippe Sosoe. On the chemical distance in critical percolation. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207032644244050243.
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