UC Berkeley

A dimer model is a probability distribution on the set of perfect matchings on a planar graph. In this thesis we study the two-periodic weighted dimer model on the Aztec diamond graph in what we call the mesocopic limit.

In the thermodynamic limit when the size of the graph goes to infinity while edge weights are fixed, the model exhibits three different regions characterized by the rate of decrease of correlation functions. At the center is the ordered region, where two-point correlation functions between dimers decrease exponentially. We show that in the mesoscopic scaling limit, when weights scale in the thermodynamic limit such that the size of the ordered region is of the same order as the correlation length inside the ordered region, fluctuations of the correlation functions are described by a new process. We compute asymptotics of the inverse Kasteleyn matrix for vertices in a local neighborhood in this mesoscopic limit, and use this to find the one-point correlation functions. We then conclude with an experimental study of two-point correlation functions between pairs of dimers that grow further apart in the mesoscopic limit.