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arXiv · 2106.14211

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

Lung-Chi Chen·Satoshi Handa·Yoshinori Kamijima
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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 $-1$.

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Lung-Chi Chen, Satoshi Handa, Yoshinori Kamijima. 2021-06-27. Mean-field behavior of nearest-neighbor oriented percolation on the BCC lattice above $8+1$ dimensions. https://arxiv.org/abs/2106.14211

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