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 $$-1$$ .