We develop the discrete set of Dyson-Schwinger equations for scalar fields and solve them for some cases. We show that their solutions are Gaussian in the continuum limit as expected from the theorems of Aizenman [Phys. Rev. Lett. 47, 886 (1981); Commun. Math. Phys. 86, 1 (1982)] and of Aizenman and Duminil-Copin [Ann. Math. 194, 163 (2021)] for $d\ge 4$. Extension to lower dimensionality fails, as it should, by observing that the triviality theorems used in our proof are not applicable in such cases.

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• Lattice field theory