Unimodular random one-ended planar graphs are sofic

TL;DR

The paper addresses whether unimodular random one-ended planar graphs are sofic and develops a method to realize a unimodular combinatorial embedding of such graphs into the plane using Whitney's theorem, Tutte decomposition, and edge amalgams. By constructing a unimodular embedding and leveraging known AHNR results, it shows that graphs with finite expected degree have a unimodular spanning tree in the Free Uniform Spanning Forest, which implies sofity. This approach effectively reduces the problem to embedding existence and simply connectedness, establishing a bridge between embedding theory and soficity. The findings extend the AHNR dichotomy to one-ended unimodular random planar graphs and clarify the role of spanning trees in attaining sofic approximations.

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.

Figures (2)

• Figure 1: A representation of (part of the) same infinite graph by a simply connected map (on the left) and by one that is not. The embedding on the left is unimodular (being an atomic probability measure on a single decorated graph), and we can make the one on the right unimodular by deciding for each triangle $\Delta$ indepedently how its three neighboring triangles should be mapped with regard to the two components of the sphere minus $\Delta$. After making these decisions for every triangle, the combinatorial embedding is determined up to orientation, and one can decide about the latter by a coin flip.
• Figure 2: Amalgamating two graphs. When both of them come with a combinatorial planar embedding, one obtains a combinatorial embedding of the amalgamated graph.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 5
• proof
• Remark 6
• Corollary 7
• Theorem 9: Whitney, Imrich
• Theorem 10