Volume 24, 2020
|Page(s)||341 - 373|
|Published online||22 September 2020|
Uniform LSI for the canonical ensemble on the 1D-lattice with strong, finite-range interaction
Department of Mathematics, University of California,
Los Angeles, USA.
* Corresponding author: email@example.com
Accepted: 8 January 2020
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.
Mathematics Subject Classification: 26D10 / 82B05 / 82B20
Key words: Canonical ensemble / logarithmic Sobolev inequality / Poincaré inequality / one-dimensional lattice / mixing condition / strong interaction
© EDP Sciences, SMAI 2020
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