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 $-1$.

41 pages, a lot of figures with TikZ

References in corpus (1)

• Diagrammatic bounds on the lace-expansion coefficients for oriented percolation