Arm exponents in high dimensional percolation

Kozma, Gady; Nachmias, Asaf

doi:10.1090/S0894-0347-2010-00684-4

Arm exponents in high dimensional percolation
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by Gady Kozma and Asaf Nachmias;

J. Amer. Math. Soc. 24 (2011), 375-409

DOI: https://doi.org/10.1090/S0894-0347-2010-00684-4

Published electronically: November 16, 2010

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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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