  The dimer model on a planar bipartite graph can be viewed as a random surface
measure. We study these fluctuations for a dimer model on the square grid with
two different classes of weights and provide a condition for their equivalence.
In the thermodynamic limit and scaling window, these height fluctuations are
shown to be non-Gaussian. They are also rotationally invariant for a certain
choice of weights. Finally, we show that the height difference between any two
points is Gaussian for a specific choice of weights.
