Uniform LSI for the canonical ensemble on the 1D-lattice with strong, finite-range interaction

Kwon, Younghak; Menz, Georg

ESAIM: Probability and Statistics, Tome 24 (2020), pp. 341-373

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show that the canonical ensemble (ce) satisfies a uniform logarithmic Sobolev inequality (LSI). The LSI constant is uniform in the boundary data, the external field and scales optimally in the system size. This extends a classical result of H.T. Yau from discrete to unbounded, real-valued spins. It also extends prior results of Landim et al. or Menz for unbounded, real-valued spins from absent- or weak- to strong-interaction. We deduce the LSI by combining two competing methods, the two-scale approach and the Zegarlinski method. Main ingredients are the strict convexity of the coarse-grained Hamiltonian, the equivalence of ensembles and the decay of correlations in the ce.

Classification : 26D10, 82B05, 82B20
Keywords: Canonical ensemble, logarithmic Sobolev inequality, Poincaré inequality, one-dimensional lattice, mixing condition, strong interaction

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