Abstract
We prove that the Fourier transform of the properly scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index α > 0 converges to \({e^{-C|k|^{\alpha\wedge2}}}\) for some \({C\in(0,\infty)}\) above the upper-critical dimension \({{{dc \equiv 2(\alpha \wedge 2)}}}\). This answers the open question remained in the previous paper (Chen and Sakai in Probab Theory Relat Fields 142:151–188, 2008). Moreover, we show that the constant C exhibits crossover at α = 2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.
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Chen, LC., Sakai, A. Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation. Probab. Theory Relat. Fields 145, 435–458 (2009). https://doi.org/10.1007/s00440-008-0174-6
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DOI: https://doi.org/10.1007/s00440-008-0174-6
Keywords
- Long-range oriented percolation
- Mean-field critical behavior
- Limit theorem
- Crossover phenomenon
- Lace expansion
- Fractional moments