Multirange Ising model on the square lattice

We study the Ising model on $\mathbb{Z}^{2}$ 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 $\mathbb{Z}^4$ as $m \to \infty$. 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 $\mathbb{Z}^6$. 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, Stochastic Process. Appl. {\bf 121}, 2043 (2011)].

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