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

We consider bond percolation on ({mathbb {Z}}^dtimes {mathbb {Z}}^s) where edges of ({mathbb {Z}}^d) are open with probability (p 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}).