The one-arm exponent for mean-field long-range percolation
| Autor: | Tim Hulshof |
|---|---|
| Přispěvatelé: | Stochastic Operations Research, Probability |
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Statistics and Probability
Range (particle radiation) Phase transition 82B43 Probability (math.PR) FOS: Physical sciences Percolation Mathematical Physics (math-ph) Critical exponent Combinatorics Mean field theory Mean-field behavior 60K35 Branching random walk 60K35 (primary) 82B27 82B43 (secondary) Exponent FOS: Mathematics Statistics Probability and Uncertainty Mathematics - Probability Mathematical Physics 82B27 Mathematics |
| Zdroj: | Electronic Journal of Probability, 20. Institute of Mathematical Statistics Electron. J. Probab. |
| ISSN: | 1083-6489 |
| DOI: | 10.1214/ejp.v20-3935 |
| Popis: | 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 $\alpha =4$. Comment: 28 pages, 1 figure, 1 appendix |
| Databáze: | OpenAIRE |
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