Factor-of-iid balanced orientation of non-amenable graphs

TL;DR

This paper proves that certain non-amenable, quasi-transitive, unimodular graphs with all degrees even admit a factor-of-iid balanced orientation, and extends results on perfect matchings to bipartite regular graphs.

Contribution

It extends spectral-theoretic results to Bernoulli graphings and establishes the existence of factor-of-iid balanced orientations and perfect matchings in broader classes of graphs.

Findings

Non-amenable, quasi-transitive, unimodular graphs with all degrees even have factor-of-iid balanced orientations.

Bipartite, regular graphs of any degree have factor-of-iid perfect matchings.

Generalization of Lyons and Nazarov's result beyond transitive graphs.

Abstract

We show that if a non-amenable, quasi-transitive, unimodular graph $G$ has all degrees even then it has a factor-of-iid balanced orientation, meaning each vertex has equal in- and outdegree. This result involves extending earlier spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of quasi-transitive, unimodular graphs. As a consequence, we also obtain that when $G$ is regular (of either odd or even degree) and bipartite, it has a factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov beyond transitive graphs.