Arm exponents in high dimensional percolation
Probability
2009-11-05 v1 Mathematical Physics
math.MP
Abstract
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.
Keywords
Cite
@article{arxiv.0911.0871,
title = {Arm exponents in high dimensional percolation},
author = {Gady Kozma and Asaf Nachmias},
journal= {arXiv preprint arXiv:0911.0871},
year = {2009}
}
Comments
42 pages, 9 figures