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

24 pages, 1 figure. This is one of two papers that are replacing the shorter arXiv submission arXiv:2101.12577v1 Factor of iid Schreier decoration of transitive graphs

References in corpus (2)

• Sublogarithmic Distributed Algorithms for Lovász Local lemma, and the Complexity Hierarchy
• Factor maps for automorphism groups via Cayley diagrams

Cited by in corpus (1)

• Factor-of-iid Schreier decorations of lattices in Euclidean spaces