# Dimensional Crossover in Anisotropic Percolation on $Z^{d+s}$

## TL;DR

This paper investigates bond percolation on a product lattice, deriving bounds for the critical curve near the threshold, and explores the transition from lower to higher-dimensional percolation behavior.

## Contribution

It provides new bounds for the critical curve in anisotropic percolation on $ ext{Z}^{d+s}$, elucidating the dimensional crossover phenomenon.

## Key findings

- Bounds for the critical curve in $(p, q)$ near $p_c( ext{Z}^d)$
- Analysis of the dimensional crossover from $ ext{Z}^d$ to $ ext{Z}^{d+s}$
- Insights into phase transition behavior in anisotropic percolation

## Abstract

We consider bond percolation on $\Z^d\times \Z^s$ where edges of $\Z^d$ are open with probability $p<p_c(\Z^d)$ and edges of $\Z^s$ are open with probability $q$, independently of all others. We obtain bounds for the critical curve in $(p, q)$, with $p$ close to the critical threshold $p_c(\Z^d)$. The results are related to the so-called dimensional crossover from $\Z^d$ to $\Z^{d+s}$.