The tortuosity of occupied crossings of a box in critical percolation |
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Authors: | Harry Kesten Yu Zhang |
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Institution: | (1) Department of Mathematics, Cornell University, 14853 Ithaca, New York;(2) Department of Mathematics, University of Colorado, 80933 Colorado Springs, Colorado |
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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
D
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 length(p) satisfies (p)(p
c}–p)
– for some <1, then with a probability tending to 1, ¦s
n¦>/C
1
n
1/(logn)–(1–)/. The assumption (p)C
3(p
c–p)– with <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
– for somec, >0. We also obtain a result about the length of optimal paths in first-passage percolation. |
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Keywords: | Critical percolation chemical distance tortuosity occupied crossings of a box lowest crossing of a square |
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