Mean-field behavior of Nearest-Neighbor Oriented Percolation on the BCC Lattice Above 8 + 1 Dimensions

Abstract

In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the d-dimensional body-centered cubic (BCC) lattice \({\mathbb {L}^d}\) and the set of non-negative integers \({{\mathbb {Z}}_+}\). Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on \({\mathbb {L}^d} \times {{\mathbb {Z}}_+}\) in all dimensions \(d\ge 9\). As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier–Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang’s bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value \(-1\).