Arm exponents in high dimensional percolation

Details Arm exponents in high dimensional percolation Arm exponents in high dimensional percolation

2009-11-04

arxiv.org/abs/0911.0871v1

Mathematics

Probability

42 pages, 9 figures

Scientific paper

We study the probability that the origin is connected to the sphere of radius r (an arm event) in critical percolation in high dimensions, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We prove that this probability decays like 1/r^2. Furthermore, we show that the probability of having k disjoint arms to distance r emanating from the vicinity of the origin is 1/r^2k.

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