Unimodular random planar graphs are sofic
Ádám Timár · arXiv (Cornell University) · 2019
We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it is sofic, that is, there is a sequence of finite graphs that converges to it in the local weak sense. First we prove that every unimodular random graph has a unimodular decomposition into finite or 1-ended subgraphs, connected by finite sets of edges in a tree-like fashion. This reduces the problem of soficity to the one-ended case. Then we show that every unimodular planar graph has a unimodular combinatorial embedding in the plane. The one-ended case then follows by a theorem of Angel, Hutchcroft, Nachmias and Ray \cite{AHNR}, who showed that every {\it simply connected} unimodular random planar {\it map} of finite expected degree is sofic. Our unimodular embedding also implies that all the dichotomy results of \cite{AHNR} about unimodular maps extend to unimodular planar graphs.