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

We consider bond percolation on ${\mathbb{Z}}^d\times {\mathbb{Z}}^s$ where edges of ${\mathbb{Z}}^d$ are open with probability $p<p_c({\mathbb{Z}}^d)$ and edges of ${\mathbb{Z}}^s$ 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 $p_c({\mathbb{Z}}^d)$. The results are related to the so-called dimensional crossover from ${\mathbb{Z}}^d$ to ${\mathbb{Z}}^{d+s}$.