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2015 The one-arm exponent for mean-field long-range percolation
Tim Hulshof
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Electron. J. Probab. 20: 1-26 (2015). DOI: 10.1214/EJP.v20-3935

Abstract

Consider a long-range percolation model on $\mathbb{Z}^d$ where the probability that an edge $\{x,y\} \in \mathbb{Z}^d \times \mathbb{Z}^d$ is open is proportional to $\|x-y\|_2^{-d \alpha}$ for some $\alpha \gt 0$ and where $d \gt 3 \min\{2,\alpha\}$. We prove that in this case the one-arm exponent equals $\min\{4,\alpha\}/2$. We also prove that the maximal displacement for critical branching random walk scales with the same exponent. This establishes that both models undergo a phase transition in the parameter $\alpha$ when $\alpha =4$.

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Tim Hulshof. "The one-arm exponent for mean-field long-range percolation." Electron. J. Probab. 20 1 - 26, 2015. https://doi.org/10.1214/EJP.v20-3935

Information

Accepted: 3 November 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1328.60214
MathSciNet: MR3418547
Digital Object Identifier: 10.1214/EJP.v20-3935

Subjects:
Primary: 60K35
Secondary: 82B27, 82B43

JOURNAL ARTICLE
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