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The One-arm Exponent for Mean-field Long-range Percolation by Hulshof, Tim is a scholarly article available to read on EtoBox.

What is The One-arm Exponent for Mean-field Long-range Percolation about?

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 $\alpha >0$ and where $d > 3 \min\{2,\alpha\}$. 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 $\alpha =4$.

Author

Hulshof, Tim

Published

2014

Language

EN