Abstract
We consider the length of an occupied crossing of a box of size [0,n]×[0, 3n]D−1 (in the short direction) in standard (Bernoulli) bond percolation on ℤD at criticality. Let ¦s n¦ be the length of the shortest such crossing. It is believed that ¦s n¦ ≈1+c in some sense for somec>0. Here we show that if the correlation lengthξ(p) satisfies ξ(p)</(p c}−p) −ν for some ν<1, then with a probability tending to 1, ¦s n¦>/C 1 n 1/ν(logn)−(1−ν)/ν. The assumption ξ(p)⩽C 3(p c−p)−ν with ν<1 has been rigorously established(1,2) for largeD, but cannot hold(3) forD=2. In the latter case, let ¦l n¦ be the length of the lowest occupied crossing of the square [0,n]2. We outline a proof ofP pc(¦ln¦ ⩽n 1+c)⩽n −α for somec, α>0. We also obtain a result about the length of optimal paths in first-passage percolation.
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Kesten, H., Zhang, Y. The tortuosity of occupied crossings of a box in critical percolation. J Stat Phys 70, 599–611 (1993). https://doi.org/10.1007/BF01053586
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DOI: https://doi.org/10.1007/BF01053586