Invariant random perfect matchings in Cayley graphs

Abstract

We prove that every non-amenable Cayley graph admits a factor of IID perfect matching. We also show that any connected $d$-regular vertex transitive graph admits a perfect matching. The two results together imply that every Cayley graph admits an invariant random perfect matching.

A key step in the proof is a result on graphings that also applies to finite graphs. The finite version says that for any partial matching of a finite regular graph that is a good expander, one can always find an augmenting path whose length is poly-logarithmic in one over the ratio of unmatched vertices.