Probability Theory and Related Fields

, Volume 145, Issue 3, pp 435–458

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

Article

DOI: 10.1007/s00440-008-0174-6

Cite this article as:
Chen, LC. & Sakai, A. Probab. Theory Relat. Fields (2009) 145: 435. doi:10.1007/s00440-008-0174-6

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.

Keywords

Long-range oriented percolation Mean-field critical behavior Limit theorem Crossover phenomenon Lace expansion Fractional moments 

Mathematics Subject Classification (2000)

60K35 82B27 

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsFu-Jen Catholic UniversityHsinchuang, Taipei HsienTaiwan
  2. 2.Creative Research Initiative “Sousei”Hokkaido UniversitySapporoJapan