Multirange Ising model on the square lattice

Abstract

We study the Ising model on Z 2 and show, via numerical simulation, that allowing interactions between spins separated by distances 1 and m (two ranges), the critical temperature, T c (m), converges monotonically to the critical temperature of the Ising model on Z 4 as m → ∞. Only interactions between spins located in directions parallel to each coordinate axis are considered. We also simulated the model with interactions between spins at distances of 1, m and u (three ranges), with u a multiple of m; in this case our results indicate that T c (m, u) converges to the critical temperature of the model on Z 6 . For percolation, analogous results were proven for the critical probability p c [B.