English

Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures

Probability 2026-03-27 v1 Mathematical Physics math.MP

Abstract

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 Rd\mathbb{R}^d is not exponentially fast.

Keywords

Cite

@article{arxiv.2603.25479,
  title  = {Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures},
  author = {Yannic Steenbeck},
  journal= {arXiv preprint arXiv:2603.25479},
  year   = {2026}
}