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.

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