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

TL;DR

This paper establishes the asymptotic behavior of the Fourier transform of the two-point function in long-range oriented percolation, revealing a crossover at  and advancing understanding of critical phenomena in high dimensions.

Contribution

It proves the convergence of the scaled two-point function's Fourier transform to a specific exponential form and introduces a new method for estimating fractional moments in lace-expansion analysis.

Findings

Fourier transform converges to e^{-C|k|^{ ext{wedge}2}} for <

Constant C shows crossover at =2 due to path interactions

Results hold above the upper-critical dimension 2( ext{wedge}2)

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 \alpha>0 converges to e^{-C|k|^{\alpha\wedge2}} for some C\in(0,\infty) above the upper-critical dimension 2(\alpha\wedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. Moreover, we show that the constant C exhibits crossover at \alpha=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.