# Central limit theorem for random surfaces

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 $R^{2}$

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