Multirange Ising model on the square lattice

We study the Ising model on the square lattice (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. N. B. de Lima, R. P. Sanchis, and R. W. C. Silva, Stochast. Process. Appl. 121, 2043 (2011)STOPB70304-414910.1016/j.spa.2011.05.009].