Factor-of-Iid Balanced Orientation of Non-Amenable Graphs

Bencs, Ferenc and Hrušková, Aranka and Tóth, László Márton (2024) Factor-of-Iid Balanced Orientation of Non-Amenable Graphs. EUROPEAN JOURNAL OF COMBINATORICS, 115. No.-103784. ISSN 0195-6698 (print); 1095-9971 (online)

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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.

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