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.
Funding Statement
F. B. has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n∘ 617747; The research was partially supported by the MTA Rényi Institute Lendület Limits of Structures Research Group. A. M. was partially supported by the ERC Consolidator Grant 648017.
Acknowledgments
The authors are grateful to Miklós Abért, Péter Csikvári and the anonymous referee for their comments on the manuscript.
Citation
Ferenc Bencs. András Mészáros. "Atoms of the matching measure." Electron. J. Probab. 27 1 - 38, 2022. https://doi.org/10.1214/22-EJP809
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