Abstract
We prove that the absolute spectral gap of any monotone Markov chain coincides with its optimal Ollivier–Ricci curvature, where the word optimal refers to the choice of the underlying metric. Moreover, we provide a new expression in terms of local variations of increasing functions, which has several practical advantages over the traditional variational formulation using the Dirichlet form. As an illustration, we explicitly determine the optimal curvature and spectral gap of the nonconservative exclusion process with heterogeneous reservoir densities on any network, despite the lack of reversibility.
Funding Statement
This work was partially supported by Institut Universitaire de France.
Acknowledgments
The author warmly thanks Pietro Caputo and Max Fathi for suggesting several interesting possible applications, which will be investigated in the near future. Thanks are also due to Alderic Joulin, Florentin Münch and Sam Power for providing relevant references.
Citation
Justin Salez. "Spectral gap and curvature of monotone Markov chains." Ann. Probab. 52 (3) 1153 - 1161, May 2024. https://doi.org/10.1214/24-AOP1688
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