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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 d4. Extension to lower dimensionality fails, as it should, by observing that the triviality theorems used in our proof are not applicable in such cases.
Marco Frasca (Thu,) studied this question.