Local Statistics of Realizable Vertex Models

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.

References in corpus (3)

• Local statistics of lattice dimers
• An introduction to the dimer model
• $0\nu \beta \beta$ nuclear matrix elements and the occupancy of individual orbits

Cited by in corpus (9)

• Counting self-avoiding walks
• Critical surface of the hexagonal polygon model
• Self-avoiding walks and the Fisher transformation
• Constrained percolation in two dimensions
• Uniqueness of Infinite Homogeneous Clusters in 1-2 Model
• Particle Systems arising from an anti-ferromagnetic Ising model
• Matchings on Random Regular Hypergraphs
• Constrained percolation, Ising model and XOR Ising model on planar lattices
• Mixing time of Markov chains for the 1-2 model