Description
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. One component of the proof is 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 can also be represented in the plane in a unimodular way. More precisely, it has a unimodular planar combinatorial embedding. The one-ended case then follows by a theorem of Angel, Hutchcroft, Nachmias and Ray. Our unimodular embedding also implies that a bunch of dichotomy results of the above authors about unimodular planar maps extend to unimodular planar graphs.