Hyperbolic and Parabolic Unimodular Random Maps

Abstract

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.

Counterexamples for percolation on unimodular random graphs

“…Then for any γ ∈ Γ, the image of γ on the partition has the same law on the partition. (A partition is described as a subset of V (S) 2 , with the product σ-algebra.) We define a tree D, such that shows the subdivision of one of the two classes appearing in (1) classes into four classes as occurs in V 2 (S).…”

“…The sketch of the argument is as follows: It is easily verified that G is invariantly amenable (see e.g. [1, Section 8] and [2]), so that p u (G) = p c (G) = q by [1,Corollary 6.11,8.13]. Thus, for every p > q, every type-1 copy of G in H contains a unique infinite open cluster almost surely.…”

The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set

“…Another corollary of Theorem 1.1 is that although the FUSF might be connected in every quasi-transitive (or more generally, unimodular random) planar graph (see [AHNR18] for a large subclass), this for sure cannot be extended from planar graphs to an arbitrary minor-closed family. This also means that a positive answer to [Tim19+, Question 8], extending treeability and soficity of unimodular random graphs from the planar case to graphs with arbitrary excluded minors, cannot be done via the strategy of [AHNR18], using the FUSF.…”

The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set

“…Another corollary of Theorem 1.1 is that although the FUSF might be connected in every quasi-transitive (or more generally, unimodular random) planar graph (see [AHNR18] for a large subclass), this for sure cannot be extended from planar graphs to an arbitrary minor-closed family. This also means that a positive answer to [Tim20, Question 8], extending treeability and soficity of unimodular random graphs from the planar case to graphs with arbitrary excluded minors, cannot be done via the strategy of [AHNR18], using the FUSF.…”