Critical behavior and the limit distribution for long-range oriented percolation. I

Abstract

We consider oriented percolation on Zd × Z+ whose bond-occupation probability is pD( · ), where p is the percolation parameter and D is a probability distribution on Zd . Suppose that D(x) decays as |x|−d−α for some α > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension dc = 2(α ∧2).We also show that, for every k, the Fourier transform of the normalized two-point function at time n,with a proper spatial scaling, has a convergent subsequence to e−c|k|α∧2 for some c > 0

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

“…This answers the open question remained in the previous paper [1]. Moreover, we show that the constant C exhibits crossover at α = 2, which is a result of interactions among occupied paths.…”

“…Next, we consider the integral ofÎ 1 . In fact, we only need consider the contribution from |∆ uφp (l, e iθ )| inŶ u (l, e iθ ) of (4.17), because the contribution from the other term inŶ u (l, e iθ ) can be estimated similarly to the integral ofÎ 2 , as explained above.…”

“…Finally, we discuss the integral ofÎ 4 . We only need consider the contribution from .20), since the contributions from the other combinations are bounded similarly to the integrals ofÎ 1 …”

Critical two-point functions for long-range statistical-mechanical models in high dimensions

“…In this paper, we investigate long-range SAW, percolation and the Ising model on Z d defined by power-law decaying pair potentials of the form D(x) ≍ |x| −d−α with α > 0. For example, as in [9,10], we can consider the following uniformly spread-out long-range D with parameter L ∈ [1, ∞):…”

The survival probability and r-point functions in high dimensions

“…In the literature, long-range models have attracted considerable attention. See [7], [8], [9] for results on long-range oriented percolation, [26] for long-range self-avoiding walk, and [27] for percolation, selfavoiding walk and the Ising model. In long-range models, the random walk step distribution D has infinite variance.…”