  Here we study the two-periodic weighted dimer model on the Aztec diamond
graph. In the thermodynamic limit when the size of the graph goes to infinity
while weights are fixed, the model develops a limit shape with frozen regions
near corners, a flat ``diamond'' in the center with a noncritical (ordered)
phase, and a disordered phase separating this diamond and the frozen phase. We
show that in the mesoscopic scaling limit, when weights scale in the
thermodynamic limit such that the size of the ``flat diamond'' is of the same
order as the correlation length inside the diamond, fluctuations of the height
function are described by a new process. We compute asymptotics of the inverse
Kasteleyn matrix for vertices in a local neighborhood in this mesoscopic limit.
