Mean-field behavior of nearest-neighbor oriented percolation on the BCC lattice above $8+1$ dimensions

TL;DR

This paper proves the infrared bound for nearest-neighbor oriented percolation on the BCC lattice in dimensions nine and higher, using Fourier analysis and lace-expansion techniques to handle complex-valued functions.

Contribution

It establishes the infrared bound for oriented percolation on the BCC lattice in high dimensions, extending previous results to include complex analysis due to time-orientation.

Findings

Infrared bound holds for dimensions d ≥ 9 on BCC lattice.

Fourier-Laplace analysis resolves issues with complex-valued functions.

Key properties of the Green function enable upper bounds in lace-expansion.

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 $d\geq 9$. 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 $-1$.