# Exactly solved models on planar graphs with vertices in $\mathbb{Z}^3$

## TL;DR

This paper extends exactly solvable edge interaction models from the square lattice to more general planar graphs in rac{1}{3}, using local deformations and star-triangle relations, preserving partition function invariance.

## Contribution

It introduces a generalized Z-invariance framework for models on planar graphs with oriented rapidity lines, expanding the applicability of integrable models.

## Key findings

- Partition function invariance under local deformations
- Extension of Z-invariance to oriented rapidity lines
- Classical Z-invariance in the quasi-classical limit

## Abstract

It is shown how exactly solved edge interaction models on the square lattice, may be extended onto more general planar graphs, with edges connecting a subset of next nearest neighbour vertices of $\mathbb{Z}^3$. This is done by using local deformations of the square lattice, that arise through the use of the star-triangle relation. Similar to Baxter's Z-invariance property, these local deformations leave the partition function invariant up to some simple factors coming from the star-triangle relation. The deformations used here extend the usual formulation of Z-invariance, by requiring the introduction of oriented rapidity lines which form directed closed paths in the rapidity graph of the model. The quasi-classical limit is also considered, in which case the deformations imply a classical Z-invariance property, as well as a related local closure relation, for the action functional of a system of classical discrete Laplace equations.