Unimodular random one-ended planar graphs are sofic

Timár, Ádám (2023) Unimodular random one-ended planar graphs are sofic. COMBINATORICS PROBABILITY & COMPUTING, 32 (6). pp. 851-858. ISSN 0963-5483

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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. Our unimodular embedding also implies that all the dichotomy results of about unimodular maps extend in the oneended case to unimodular random planar graphs.

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