Unimodular random one-ended planar graphs are sofic

TL;DR

This paper proves that unimodular random one-ended planar graphs with finite expected degree can be embedded into the plane in a unimodular way, extending existing dichotomy results to this class of graphs.

Contribution

It establishes the existence of a unimodular embedding for one-ended planar graphs, linking unimodularity with planarity and extending prior dichotomy results.

Findings

Unimodular random planar graphs with finite expected degree admit unimodular embeddings.

The results extend dichotomy theorems to one-ended unimodular planar graphs.

The embedding facilitates analysis of unimodular maps in planar graphs.

Abstract

We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended case to unimodular random planar graphs.