Local correlation functions of the two-periodic weighted Aztec diamond in mesoscopic limit

TL;DR

This paper investigates the mesoscopic limit of the two-periodic weighted Aztec diamond model, revealing a new fluctuation process for the height function and providing asymptotic analysis of the inverse Kasteleyn matrix.

Contribution

It introduces a new fluctuation process for the height function in the mesoscopic scaling limit of the model and computes the asymptotics of the inverse Kasteleyn matrix in this regime.

Findings

Identification of a new fluctuation process in the mesoscopic limit

Asymptotic formulas for the inverse Kasteleyn matrix

Description of phase separation in the model

Abstract

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.