A note on the dimensional crossover critical exponent

TL;DR

This paper investigates anisotropic bond percolation on a product lattice, establishing bounds on the crossover critical exponent and proving its value for high dimensions, advancing understanding of phase transitions in anisotropic systems.

Contribution

It provides new bounds on the dimensional crossover critical exponent and confirms its value as 1 for dimensions d ≥ 11.

Findings

Percolation occurs for q ≥ 8d²(p_c(Z^d) - p)

The crossover critical exponent, if it exists, is greater than 1

For d ≥ 11, the crossover critical exponent equals 1

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 $q\geq 8d^2(p_c(\mathbb{Z}^d)-p)$. 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 $d\geq 11$, the crossover critical exponent exists and equals 1.