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Arm exponents in high dimensional percolation

Authors: Gady Kozma and Asaf Nachmias
Journal: J. Amer. Math. Soc. 24 (2011), 375-409
MSC (2010): Primary 60K35, 82B43
Published electronically: November 16, 2010
MathSciNet review: 2748397
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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 $ r^{-2}$. Furthermore, we show that the probability of having $ \ell$ disjoint arms to distance $ r$ emanating from the vicinity of the origin is $ r^{-2\ell}$.

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Additional Information

Gady Kozma
Affiliation: The Weizmann Institute of Science, Rehovot POB 76100, Israel

Asaf Nachmias
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Received by editor(s): November 4, 2009
Received by editor(s) in revised form: July 21, 2010
Published electronically: November 16, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.