Local Statistics of Realizable Vertex Models

TL;DR

This paper analyzes planar vertex models, expressing their local statistics via dimer models using a holographic algorithm, and provides explicit formulas for free energy and local statistics on infinite graphs, with simulation examples.

Contribution

It introduces a method to relate vertex models to dimer models through a generalized holographic algorithm, enabling explicit calculations of local statistics and free energy.

Findings

Derived explicit integral formulas for free energy.

Expressed local statistics as a linear combination of dimer model statistics.

Simulated the 1-2 model using Glauber dynamics.

Abstract

We study planar "vertex" models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including dimer model, and 1-2 model, which we will define. We express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding Fisher graph, with the help of a generalized holographic algorithm. Using an $n\times n$ torus to approximate the periodic infinite graph, we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi-periodic graph. As an example, we simulate the 1-2 model by the technique of Glauber dynamics.