Arm exponents in high dimensional percolation

2009/11/04 by Kozma, Gady, Nachmias, Asaf · 6 citations
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.0911.0871

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/r². Furthermore, we show that the probability of having k disjoint arms to distance r emanating from the vicinity of the origin is 1/r²k.

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• Dimension dependence of critical phenomena in long-range percolation
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• Limiting distribution of the chemical distance in high dimensional critical percolation
• Decay of connection probability in high-dimensional continuum percolation

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