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Suppose I have developed a new Monte Carlo method, and I plan to test this method on studying the magnetization of a 3D Ising model at some non-zero temperature $T$. The coupling is nearest neighbor, with the coupling strength $J_{ij}$ being random at each site. What is the standard way to evaluate the quality of the simulations without an exact solution of the magnetization for comparison, besides using analytical approximations such as the mean-field method or high temperature expansion? In lower dimensions (such as 2D), is there a way to "plant" an Ising instance such that the magnetization can be determined exactly besides the case of the trivial ferromagnetic instance?

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    $\begingroup$ computational science SE might be a better home for this question $\endgroup$ – By Symmetry May 26 at 12:23
  • $\begingroup$ Related question: physics.stackexchange.com/q/476953 $\endgroup$ – tpg2114 May 26 at 13:13
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    $\begingroup$ @BySymmetry Taking into account that Computationl Physics is not OT on physics.SE, this is more a question about Computational Physics (or just Physics) than Computational Science. The reason for my claim is that this is not a question about a specific computational algorithm, but it touches the important question "how can we know that numerical results for a specific model of a physical system can be trusted?". $\endgroup$ – GiorgioP May 26 at 14:53
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Soon a later, computational methods bring computational physicists in unknown lands never visited before. So, at some point one has to learn how to trust numerical results without the guidance of known cases.

Of course, before trusting a numerical code on new systems, people extensively test it on well known cases, by comparison with i) exact results (if they exist) ii) perturbative results, iii) approximate theories, iv) other simulations.

In your case, you should be able to use as a test some of many Monte Carlo results for a three-dimensional random first-neighbor Ising model. A quick search on Google Scholar should provide enough references. Once you are sure you can reproduce literature results, you can move to your own specific model with some confidence

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I don't know whether this would be useful to test your algorithm, but there is a bond-disordered version of the 2d Ising model for which some quantities have been computed exactly by McCoy and Wu.

The model and results (there may be more recent ones) are described in Chapters XIV and XV of their book. Their model has all "horizontal" coupling constants equal, but the "vertical" coupling constants are only equal in each column and sampled independently for each column.

Among the quantities they computed are the specific heat, the average boundary magnetization and the average 2-point function.

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