Atoms of the matching measure

TL;DR

This paper investigates the properties of the matching measure in infinite graphs, proving it has no atoms in vertex-transitive cases and finitely many in certain non-amenable graphs, extending finite graph results.

Contribution

It extends the understanding of matching measures and Gallai-Edmonds decomposition from finite to infinite graphs, including new results for non-amenable unimodular random graphs.

Findings

Matching measure of vertex-transitive graphs has no atoms.

Finitely many atoms in non-amenable unimodular random graphs.

Gallai-Edmonds decomposition is compatible with the monomer-dimer model.

Abstract

We prove that the matching measure of an infinite vertex-transitive connected graph has no atoms. Generalizing the results of Salez, we show that for an ergodic non-amenable unimodular random rooted graph with uniformly bounded degrees, the matching measure has only finitely many atoms. Ku and Chen proved the analogue of the Gallai-Edmonds structure theorem for non-zero roots of the matching polynomial for finite graphs. We extend their results for infinite graphs. We also show that the corresponding Gallai-Edmonds decomposition is compatible with the zero temperature monomer-dimer model.