Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918411120934912 |
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| author | Steenbeck, Yannic |
| author_facet | Steenbeck, Yannic |
| contents | We prove that there is only one translation-invariant Gibbsian point process w.r.t. to a chosen interaction if any of them satisfies a certain bound related to concentration-of-measure. This concentration-of-measure bound is e.g. fulfilled if a corresponding modified logarithmic Sobolev inequality holds. In particular, for natural examples with non-uniqueness regimes, a modified logarithmic Sobolev inequality cannot be satisfied. Therefore, in these situations, the free-energy dissipation in related continuous-time birth-and-death dynamics in $\mathbb{R}^d$ is not exponentially fast. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_25479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures Steenbeck, Yannic Probability Mathematical Physics 82C21, 82B21, 60F10, 46E35 (Primary) 60G55, 60E15, 60H07, 39B62, 60K35, 60J25 (Secondary) We prove that there is only one translation-invariant Gibbsian point process w.r.t. to a chosen interaction if any of them satisfies a certain bound related to concentration-of-measure. This concentration-of-measure bound is e.g. fulfilled if a corresponding modified logarithmic Sobolev inequality holds. In particular, for natural examples with non-uniqueness regimes, a modified logarithmic Sobolev inequality cannot be satisfied. Therefore, in these situations, the free-energy dissipation in related continuous-time birth-and-death dynamics in $\mathbb{R}^d$ is not exponentially fast. |
| title | Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures |
| topic | Probability Mathematical Physics 82C21, 82B21, 60F10, 46E35 (Primary) 60G55, 60E15, 60H07, 39B62, 60K35, 60J25 (Secondary) |
| url | https://arxiv.org/abs/2603.25479 |