A note on the dimensional crossover critical exponent

Abstract

We consider independent anisotropic bond percolation on ${\mathbb{Z}}^d\times {\mathbb{Z}}^s$ where edges parallel to ${\mathbb{Z}}^d$ are open with probability $p<p_c({\mathbb{Z}}^d)$ and edges parallel to ${\mathbb{Z}}^s$ are open with probability $q$, independently of all others. We prove that percolation occurs for $q\ge 8d^2(p_c({\mathbb{Z}}^d)-p)$. This fact implies that the so-called Dimensional Crossover critical exponent, if it exists, is greater than 1. In particular, using known results, we conclude the proof that, for $d\ge 11$, the crossover critical exponent exists and equals 1.

9 pages. arXiv admin note: text overlap with arXiv:1706.07495