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I have a question about the phase transitions in the Ising model in the presence of a (constant) external magnetic field. I will state the question on $\mathbb Z^2$ for simplicity. A definition of the model is here.

It seems to be well known in the mathematical physics world that in the case of constant non-zero external magnetic field ($h_j= h>0\text{ for all $j\in\mathbb Z^2$ and }\mu>0$ in the notation of the Wikipedia article), there is no phase transition. That is, for all inverse temperatures $\beta$, the thermodynamic limits of the finite volume Gibbs measures with the all $+1$ boundary conditions and the all $-1$ boundary conditions are equal.

As far as I can tell, proofs of this are based on the Lee-Yang theorem (Messager, Miracle-Sole and Pfister) or on the GHS (Griffiths Hurst and Sherman) inequality. They are therefore based on the analytic properties of thermodynamic functions obtained by taking the limit over increasing domains.

Are there any elementary combinatorial proofs known? While hard to define exactly what I mean, what I have in mind is proofs based on estimating probabilities of configurations such as the proof of the existence of phase transition for Ising model in the absence of external field.
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  • $\begingroup$ By "the Ising model" I assume you mean the one dimensional Ising model? Also, (classical) phase transitions are defined (at least physically) as a discontinuity in the order parameter (or it's first derivative) in the system being described. For the Ising model, the order parameter is the magnetization, which is dependent upon the magnetic field strength. At finite temperature in the 1-D Ising model the magnetization and its derivative vary continuously with respect to magnetic field strength, however a discontinuity can be introduced at T=0. $\endgroup$
    – Matthew Titsworth
    Commented Dec 21, 2013 at 17:21
  • $\begingroup$ I mean the Ising model on a general graph, but the question is specifically about the Ising model on $\mathbb Z^2$. By phase transition, I mean the existence of more than one Gibbs measure. $\endgroup$
    – Anthony Quas
    Commented Dec 21, 2013 at 17:26
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    $\begingroup$ Onsager's solution is for the case with zero field though, which falls outside of your question. My apologies. Baxter's book may still be useful though. $\endgroup$
    – Matthew Titsworth
    Commented Dec 21, 2013 at 17:46
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    $\begingroup$ I haven't really thought about it, but one should be able to do it using Aizenman's random current representation. For example, it is quite easy to prove exponential decay of the truncated 2-point function at non-zero field using this representation, and a variant should be enough to establish uniqueness. The argument for the decay of correlation can be found, for example, here (see p.13) : iew3.technion.ac.il/~ieioffe/Papers/LNM.pdf . $\endgroup$
    – Yvan Velenik
    Commented Jan 13, 2014 at 12:45
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    $\begingroup$ Simple answer: phase transitions in an external field are possible on trees ( and more generally on non-amenable graphs). See for example Georgii’s book. $\endgroup$
    – Aernout van Enter
    Commented Nov 3, 2021 at 20:39

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