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 $R^{2}$

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