It's expected that the pressure of the Ising model on a $d$-dimensional discrete torus with side length $L$ converges to the mean-field Ising model pressure as the dimension $d$ goes to infinity. Is there a rigorous proof for this fact? Intuitively this is indeed true. Thanks for any references.

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    $\begingroup$ What do you mean by pressure in the Ising model? $\endgroup$
    – Zack
    Commented Aug 6, 2021 at 21:07
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    $\begingroup$ I don't think the torus has anything to do with this. It is just conventional to set the Ising model on a torus as a convenient way of choosing boundary conditions. In the thermodynamic limit, these boundary conditions should be irrelevant to the behavior of the system. So the question is really: Does the Ising model behave according to the predictions of mean field theory as the dimensionality $d$ goes to infinity? $\endgroup$
    – Buzz
    Commented Aug 7, 2021 at 1:32
  • $\begingroup$ @Zack, the log partition function, also known as free energy $\endgroup$
    –  apple
    Commented Aug 7, 2021 at 4:29
  • $\begingroup$ @Buzz, Yes, I will say that's exactly the motivation for this question. The torus setting is just for convenience, directly working on lattice could be technical. $\endgroup$
    –  apple
    Commented Aug 7, 2021 at 4:32

1 Answer 1


Yes, this was first proved (as far as I know) by Colin J. Thompson in his paper Ising model in the high density limit, published in Communication in Mathematical Physics 36(4), 255-262, 1974.

There are also similar results for the magnetization, as well as inequalities relating these quantities on finite-dimensional lattices and in the mean-field approximation. You can also look at the (brief) discussion and the references given in Sections 2.5.4 and 3.10.2 in our book for additional information on this topic.

There is also a discussion (with some proofs) of such issues in Section II.14 of Simon's book.


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