Anisotropic Ising Model in $$d+s$$ Dimensions

Abstract

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.