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

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∧2 for some C∈(0,∞) above the upper-critical dimension dc≡2(α∧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.