Abstract
We consider the disordered monomer-dimer model on cylinder graphs \({\mathcal {G}}_n\), i.e., graphs given by the Cartesian product of the line graph on n vertices, and a deterministic finite graph. The edges carry i.i.d. random weights, and the vertices also have i.i.d. random weights, not necessarily from the same distribution. Given the random weights, we define a Gibbs measure on the space of monomer-dimer configurations on \({\mathcal {G}}_n\). We show that the associated free energy converges to a limit and, with suitable scaling and centering, satisfies a Gaussian central limit theorem. We also show that the number of monomers in a typical configuration satisfies a law of large numbers and a Gaussian central limit theorem with appropriate centering and scaling. Finally, for an appropriate height function associated with a matching, we show convergence to a limiting function and prove the Brownian motion limit around the limiting height function in the sense of finite-dimensional distributional convergence.
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Acknowledgements
We want to thank the anonymous referee for providing helpful comments and suggestions to improve the clarity and presentation of the paper. We thank Felix Christian Clemen, Gayana Jayasinghe, Grigory Terlov, and Qiang Wu for many enlightening discussions.
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Communicated by Abhishek Dhar.
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Dey, P.S., Krishnan, K. Disordered Monomer-Dimer Model on Cylinder Graphs. J Stat Phys 190, 146 (2023). https://doi.org/10.1007/s10955-023-03159-7
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DOI: https://doi.org/10.1007/s10955-023-03159-7