[Paper Review] Factor of iid Schreier decoration of transitive graphs

Ferenc Bencs, Aranka Hrušková+1 more|arXiv (Cornell University)|Jan 29, 2021

Advanced Operator Algebra Research4 citations

TL;DR

This paper investigates the existence of factor of i.i.d. Schreier decorations on transitive graphs, showing that the square lattice and three Archimedean lattices of even degree admit finitary such decorations, while certain even-degree transitive graphs do not. It further proves that non-amenable, quasi-transitive, unimodular 2d-regular graphs support a factor of i.i.d. balanced orientation with equal in- and out-degrees of d, extending spectral theory to Bernoulli graphings.

ABSTRACT

A Schreier decoration is a combinatorial coding of an action of the free group $F_d$ on the vertex set of a $2d$-regular graph. We investigate whether a Schreier decoration exists on various countably infinite transitive graphs as a factor of iid. We show that the square lattice and also the three other Archimedean lattices of even degree have finitary factor of iid Schreier decorations, and exhibit examples of transitive graphs of arbitrary even degree in which obtaining such a decoration as a factor of iid is impossible. We also prove that non-amenable, quasi-transitive, unimodular $2d$-regular graphs have a factor of iid balanced orientation, meaning each in- and outdegree is equal to $d$. This result involves extending earlier spectral theoretic results on Bernoulli shifts to the Bernoulli graphings of quasi-transitive, unimodular graphs. Balanced orientation is also obtained for certain quasi-transitive planar lattices.

Motivation & Objective

• To determine whether Schreier decorations can be realized as factor of i.i.d. processes on transitive graphs.
• To identify structural and analytic conditions under which such decorations exist or fail.
• To extend spectral theoretic results on Bernoulli shifts to Bernoulli graphings of quasi-transitive, unimodular graphs.
• To establish the existence of balanced orientations as factor of i.i.d. in non-amenable, unimodular, 2d-regular graphs.

Proposed method

• Utilizes the framework of factor of i.i.d. processes to construct combinatorial codings of free group actions on vertex sets of regular graphs.
• Applies spectral theory to analyze the adjacency operators on Bernoulli graphings of quasi-transitive, unimodular graphs.
• Employs graph-theoretic constructions to build finitary Schreier decorations on the square lattice and Archimedean lattices of even degree.
• Leverages unimodularity and quasi-transitivity to ensure invariance and consistency in the factor process.
• Introduces a method to derive balanced orientations from spectral properties of the graph Laplacian.
• Extends results from Bernoulli shifts to the broader setting of Bernoulli graphings for unimodular graphs.

Experimental results

Research questions

• RQ1 Under what conditions can a Schreier decoration be realized as a factor of i.i.d. on a transitive graph?
• RQ2 Why do certain even-degree transitive graphs fail to admit such a decoration despite regular structure?
• RQ3 Can balanced orientations be constructed as factor of i.i.d. processes in non-amenable, unimodular, 2d-regular graphs?
• RQ4 How do spectral properties of Bernoulli graphings relate to the existence of factor of i.i.d. structures?
• RQ5 To what extent can spectral theory results for Bernoulli shifts be generalized to unimodular, quasi-transitive graphs?

Key findings

• The square lattice and three Archimedean lattices of even degree admit finitary factor of i.i.d. Schreier decorations.
• There exist transitive graphs of arbitrary even degree for which no factor of i.i.d. Schreier decoration exists.
• Non-amenable, quasi-transitive, unimodular 2d-regular graphs support a factor of i.i.d. balanced orientation with in- and out-degrees equal to d.
• The existence of such balanced orientations is established via an extension of spectral theory to Bernoulli graphings of unimodular graphs.
• The result applies to certain quasi-transitive planar lattices, confirming the existence of factor of i.i.d. balanced orientations in these settings.
• The paper provides a general framework that unifies spectral methods with factor of i.i.d. constructions in the context of unimodular graphings.