# A unimodular Liouville hyperbolic souvlaki --- an appendix to [arXiv:1603.06712]

## TL;DR

This paper modifies a previous hyperbolic graph construction to produce a unimodular random graph that is transient, hyperbolic, has no transient subtrees, and satisfies the Liouville property, advancing understanding of harmonic functions on such graphs.

## Contribution

It introduces a new unimodular random graph with specific properties, extending prior work on hyperbolic graphs and harmonic functions.

## Key findings

- Constructed a unimodular hyperbolic graph with no transient subtrees.
- Proved the graph has the Liouville property for harmonic functions.
- Extended the class of known hyperbolic graphs with these properties.

## Abstract

Carmesin, Federici, and Georgakopoulos [arXiv:1603.06712] constructed a transient hyperbolic graph that has no transient subtrees and that has the Liouville property for harmonic functions. We modify their construction to get a unimodular random graph with the same properties.