[Paper Review] Invariant embeddings of unimodular random planar graphs

Itaï Benjamini, Ádám Timár|arXiv (Cornell University)|Oct 3, 2019

Stochastic processes and statistical mechanics32 references4 citations

TL;DR

This paper establishes that an ergodic unimodular random one-ended planar graph with finite expected degree admits an isometry-invariant, locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. For nonamenable graphs, such embeddings exist in the hyperbolic plane. The results are extended to unimodular tilings, with constructions using invariant point processes (Euclidean case) and circle packings (hyperbolic case), providing a complete dichotomy for one-ended graphs.

ABSTRACT

Consider an ergodic unimodular random one-ended planar graph $\G$ of finite expected degree. We prove that it has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. By "locally finite" we mean that any bounded open set intersects finitely many embedded edges. In particular, there exist invariant embeddings in the Euclidean plane for the Uniform Infinite Planar Triangulation and for the critical Augmented Galton-Watson Tree conditioned to survive. Roughly speaking, a unimodular embedding of $\G$ is one that is jointly unimodular with $\G$ when viewed as a decoration. We show that $\G$ has a unimodular embedding in the hyperbolic plane if it is invariantly nonamenable, and it has a unimodular embedding in the Euclidean plane if and only if it is invariantly amenable. Similar claims hold for representations by tilings instead of embeddings.

Motivation & Objective

• To determine when an ergodic unimodular random one-ended planar graph with finite expected degree admits an isometry-invariant, locally finite embedding in the Euclidean or hyperbolic plane.
• To characterize the existence of unimodular embeddings and tilings by relating them to the invariant amenability of the graph.
• To provide constructive methods—via invariant point processes in the Euclidean case and circle packings in the hyperbolic case—for building such embeddings and tilings.
• To resolve the dichotomy between amenable and nonamenable graphs in the context of unimodular embeddings and tilings of the plane.
• To extend the framework to include tiling representations and clarify the role of unimodularity in preserving invariance under isometries.

Proposed method

• Construct an isometry-invariant embedding in the Euclidean plane by starting from a suitable invariant point process as the vertex set, ensuring the resulting embedded graph is unimodular.
• Use circle packing techniques to construct a unimodular embedding for invariantly nonamenable graphs in the hyperbolic plane, leveraging the rigidity of hyperbolic geometry.
• Define unimodular embeddings as those where the joint law of the graph and its embedding is unimodular, ensuring stationarity under automorphisms.
• Prove the necessity of invariant amenability for Euclidean embeddings by contradiction, using invariant random partitions of the plane into squares.
• Reduce the tiling problem to the embedding problem by selecting a random point in each tile and connecting it to neighbors, preserving invariance and local finiteness.
• Apply results from Aldous and Lyons [1] on Palm measures to show that the Palm version of an invariant embedding yields a unimodular embedding in the amenable case.

Experimental results

Research questions

• RQ1 Under what conditions does an ergodic unimodular random one-ended planar graph with finite expected degree admit an isometry-invariant, locally finite embedding in the Euclidean plane?
• RQ2 Can every invariantly nonamenable unimodular random planar graph be unimodularly embedded in the hyperbolic plane?
• RQ3 Is there a unimodular tiling representation of such graphs in the Euclidean or hyperbolic plane, and when does it exist?
• RQ4 What is the relationship between invariant amenability and the existence of unimodular embeddings or tilings in symmetric spaces like the Euclidean and hyperbolic planes?
• RQ5 How do the geometric properties of the underlying space (Euclidean vs. hyperbolic) affect the possibility of constructing unimodular embeddings?

Key findings

• An ergodic unimodular random one-ended planar graph with finite expected degree has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable.
• For invariantly nonamenable graphs, a unimodular locally finite embedding exists in the hyperbolic plane, but not in the Euclidean plane.
• The same dichotomy holds for unimodular tiling representations: amenable graphs admit such tilings in the Euclidean plane, nonamenable ones in the hyperbolic plane.
• The expected area of the tile containing the origin is finite in all unimodular tiling constructions, ensuring local finiteness.
• The proof of the 'only if' direction in the Euclidean case relies on constructing an invariant finite exhaustion via random partitions of the plane, which forces amenability.
• The construction for the amenable case uses the Palm version of an invariant embedding, which is shown to be unimodular via arguments from Aldous and Lyons [1].