Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation

2008/04/13 by Lung-Chi Chen, Akira Sakai, Chen, Lung-Chi +1
Computer Science · Mathematics · #60K35 #82B27 #Bayesian Methods and Mixture Models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0804.2039

openalex publication_date 2008/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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|^α\wedge2 for some C∈(0,∞) above the upper-critical dimension 2(α\wedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. 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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