The Height fluctuations of an off-critical dimer model on the square grid

TL;DR

This paper investigates height fluctuations in an off-critical dimer model on a square grid, revealing non-Gaussian behavior generally, rotational invariance under certain weights, and Gaussian height differences for specific weight choices.

Contribution

It provides conditions for equivalence of two weight classes and characterizes the nature of height fluctuations in the off-critical regime.

Findings

Height fluctuations are non-Gaussian in the thermodynamic limit.

Height fluctuations are rotationally invariant for certain weights.

Height difference between two points is Gaussian for specific weights.

Abstract

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.