Anisotropic Ising model in d+s dimensions

In this note, we consider the asymmetric nearest neighbor ferromagnetic Ising model on the $(d+s)$-dimensional unit cubic lattice ${\mathbb{Z}}^{d+s}$, at inverse temperature $\beta =1$ and with coupling constants $J_s>0$ and $J_d>0$ for edges of ${\mathbb{Z}}^s$ and ${\mathbb{Z}}^d$, respectively. We obtain a lower bound for the critical curve in the phase diagram of $(J_s,J_d)$. In particular, as $J_d$ approaches its critical value from below, our result is directly related to the so-called dimensional crossover phenomenon.