Entropy and curvature: beyond the Peres-Tetali conjecture
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914672454664192 |
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| author | Caputo, Pietro Münch, Florentin Salez, Justin |
| author_facet | Caputo, Pietro Münch, Florentin Salez, Justin |
| contents | We study Markov chains with non-negative sectional curvature on finite metric spaces. Neither reversibility, nor the restriction to a particular combinatorial distance are imposed. In this level of generality, we prove that a 1-step contraction in the Wasserstein distance implies a 1-step contraction in relative entropy, by the same amount. Our result substantially strengthens a recent breakthrough of the second author, and has the advantage of being applicable to arbitrary scales. This leads to a time-varying refinement of the standard Modified Log-Sobolev Inequality (MLSI), which allows us to leverage the well-acknowledged fact that curvature improves at large scales. We illustrate this principle with several applications, including birth and death chains, colored exclusion processes, permutation walks, Gibbs samplers for high-temperature spin systems, and attractive zero-range dynamics. In particular, we prove a MLSI with constant equal to the minimal rate increment for the mean-field zero-range process, thereby answering a long-standing question. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17148 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropy and curvature: beyond the Peres-Tetali conjecture Caputo, Pietro Münch, Florentin Salez, Justin Probability Combinatorics Differential Geometry 60J27, 60J10 We study Markov chains with non-negative sectional curvature on finite metric spaces. Neither reversibility, nor the restriction to a particular combinatorial distance are imposed. In this level of generality, we prove that a 1-step contraction in the Wasserstein distance implies a 1-step contraction in relative entropy, by the same amount. Our result substantially strengthens a recent breakthrough of the second author, and has the advantage of being applicable to arbitrary scales. This leads to a time-varying refinement of the standard Modified Log-Sobolev Inequality (MLSI), which allows us to leverage the well-acknowledged fact that curvature improves at large scales. We illustrate this principle with several applications, including birth and death chains, colored exclusion processes, permutation walks, Gibbs samplers for high-temperature spin systems, and attractive zero-range dynamics. In particular, we prove a MLSI with constant equal to the minimal rate increment for the mean-field zero-range process, thereby answering a long-standing question. |
| title | Entropy and curvature: beyond the Peres-Tetali conjecture |
| topic | Probability Combinatorics Differential Geometry 60J27, 60J10 |
| url | https://arxiv.org/abs/2401.17148 |