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It is mentioned in this answer Is there a program to solve The Yang–Mills Existence and Mass Gap problem similar to the Hamilton's program to solve Poincaré Conjecture? that it is an open problem to construct pure Yang-Mills theory in spacetime dimension $d=4$ on an infinite lattice. That made me wonder if there is any model of Euclidean QFT constructed on an infinite lattice for $d>3$.

To be more concrete, my question is: is there a construction of the $\phi^4$-model on an infinite lattice when $d>3$? If not, why not? In particular, is there a reason to think that such a model would be trivial? (As far as I know the triviality results for $\phi^4$ apply to the continuum limit only.) (Note: here by "construction" I mean non-perturbative mathematically rigorous construction.)

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You can find such a construction (for any dimension) in the article Statistical mechanics of systems of unbounded spins by Lebowitz and Presutti (see Theorem 4.3 there). As mentioned by Carlo, there are no grounds to believe that the theory on the lattice is Gaussian. In fact, one can prove that it is not (at least in some regimes) by showing that is has better than Gaussian tail behaviour.

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  • $\begingroup$ Thanks a lot, that's what I was looking for. I wonder if these models have one-particle states in the sense of this article: projecteuclid.org/journals/… so that a scattering matrix can be obtained. $\endgroup$
    – S.Z.
    Commented Jun 19 at 15:06
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The $\phi^4$ theory on a hypercubic lattice with $d$ space-time dimensions is "trivial" for $d\geq 4$, in the sense that it reduces to a free non-interacting theory in the limit that the lattice spacing $a$ goes to zero. A rigorous proof for $d=4$ has been published only recently (2021), by Aizenman and Duminil-Copin in Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models. A proof for $d>4$ was obtained earlier (1981), by Aizenman.

For finite $a$ interaction terms persist, see Lüscher and Weisz (1987).

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    $\begingroup$ I believe I clearly stated that I'm interested in the lattice theory, not the continuum limit. I'm well aware of the triviality result for the continuum limit and that's also mentioned in my question. $\endgroup$
    – S.Z.
    Commented Jun 18 at 17:49
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    $\begingroup$ Doesn't Aizenman & Duminil-Copin qualify as a $d=4$ construction of the $\phi^4$ theory on a lattice, even though they are interested in the limit $a\rightarrow 0$? I have added a reference on finite-$a$ effects, where interactions remain. $\endgroup$
    – Carlo Beenakker
    Commented Jun 18 at 18:03
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    $\begingroup$ Aizenman & Duminil-Copin do not construct any model in that paper, they prove that if the continuum/scaling limits of certain models exist then they are Gaussian. $\endgroup$
    – S.Z.
    Commented Jun 18 at 19:32
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    $\begingroup$ Thank you for the Lüscher and Weisz reference. So according to it there is no reason no think that the theory on infinite lattice is Gaussian. But they don't seem to provide rigorous construction on $\phi^4$ on infinite lattice. $\endgroup$
    – S.Z.
    Commented Jun 18 at 19:37

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