Unimodular random one-ended planar graphs are sofic (Q6632781)

Unimodular graphs are random rooted graphs \((G, o)\) having the property that the measure on the space of doubly-rooted graphs\N\[\N\mu := \mathbb{E}\bigg[\sum_{x\in V(G)} \delta_{(G,o,x)}(\cdot)\bigg]\N\]\Nis invariant under the permutation of the two roots. Sofic graphs are those that can be obtained as local limits of uniformly rooted finite graphs and are necessarily unimodular. The converse question ``Is every unimodular graph sofic?'' is therefore natural but still unanswered except in particular cases, as pointed out in [\textit{D. J. Aldous} and \textit{R. Lyons}, Electron. J. Probab. 12, 1454--1508 (2007; Zbl 1131.60003)].\N\NIn this paper, the author shows that unimodular random one-ended planar graphs are sofic. The key to the proof is to show that every locally finite unimodular random planar graph admits a unimodular combinatorial embedding into the plane (a random decoration of all vertices with a cyclic order on their neighbors that is compatible with an embedding into the plane, such that the graph, additionally marked with this decoration, remains unimodular). This is shown by first reducing the problem to 2-connected graphs, then to their 3-connected components, which have (if they are planar) a unique combinatorial embedding up to a reversal of orientation.\N\NAnother consequence of the main results of this article is that all the dichotomy results of Theorem 1 in [\textit{O. Angel} et al., Geom. Funct. Anal. 28, No. 4, 879--942 (2018; Zbl 1459.60018)] extend to unimodular random one-ended planar graphs.