Symmetric Vertex Models on Planar Random Graphs

TL;DR

This paper solves a symmetric 4-bond vertex model on planar random graphs, revealing its connection to the Ising model and quantum gravity, and extends known solutions to a new class of random graph models.

Contribution

It introduces a solution method for symmetric vertex models on random graphs by mapping to the Ising model, generalizing previous lattice solutions to random graph ensembles.

Findings

Vertex weights relate to Ising parameters as in honeycomb lattice models.

Partition function symmetry corresponds to a change of variables in the matrix integral.

Vertex models may aid in discretizing 2D Lorentzian quantum gravity.

Abstract

We solve a 4-(bond)-vertex model on an ensemble of 3-regular Phi3 planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent -- a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models). The relations between the vertex weights and Ising model parameters in the 4-vertex model on Phi3 graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model. Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.