The bounded 19-vertex model

TL;DR

This paper investigates the 19-vertex model in statistical mechanics with domain wall boundary conditions, revealing the emergence of limit shapes through a dynamic approach and exploring the connection between local paths and macroscopic geometry.

Contribution

It introduces a dynamic version of the 19-vertex model using minimal flip actions, demonstrating the presence of limit shapes beyond the 6-vertex model.

Findings

Limit shapes are observed for all dynamic weight values.

The existence of limit shapes is more general than the 6-vertex model.

Local path geometry relates to the macroscopic shape geometry.

Abstract

We study the 19-vertex model of Statistical Mechanics in a square with the domain wall boundary condition. Using the minimal set of generating flip actions we build a parametrized dynamic version of the model. For all observed dynamic weight values the equilibrium states exhibit clear limit shapes. Although the model in a way incorporates the 6-vertex model, the reason for the existence of the limit shape is fundamentally more general. We conclude with a further study relating the local path geometry to the macroscopic shape geometry.