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In the works of Spohn, Borodin and others, there are results on the height functions for random matrices showing they have fluctuations converging to the Gaussian free field. So I think we are nearing some statement of central limit theorem for random surfaces.

Q1: As with the original CLT, are there any first results for "Bernouli" random surfaces converging to a universal object eg. GFF? By "Bernouli" random surface I am thinking of 2d step functions going between level 1 and -1 with probability 1/2 i.e. process $(X_{i})$ and squares $[a_i,b_i]^{2}$ tiling $R^{2}$

$$P[X_i=(x,y,1), (x,y)\in [a_i,b_i]^{2}]=\frac{1}{2} \text{ and }P[X_i=(x,y,-1), (x,y)\in [a_i,b_i]^{2}]=\frac{1}{2}.$$

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I think yes, it's implied by the following (from Wikipedia https://en.m.wikipedia.org/wiki/Gaussian_free_field):

Similarly to Brownian motion, which is the scaling limit of a wide range of discrete random walk models (see Donsker's theorem), the continuum GFF is the scaling limit of not only the discrete GFF on lattices, but of many random height function models, such as the height function of uniform random planar domino tilings, see Kenyon (2001).

Note that Donsker's Theorem is a generalization of the Central Limit Theorem.

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