Equivalence of a mixing condition and the LSI in spin systems with infinite range interaction

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Henderson, Christopher¹  Menz, Georg² 

[1] Ecole Normale Super Lyon, Unite Math Pures & Appl, Lyon, France

[2] Univ Calif Los Angeles, Dept Math, Los Angeles, CA 90024 USA

关键词: Spin systems;    Log-Sobolev inequality;    Poincare inequality;    Decay of correlations;    Phase transition;    Gibbs measure;    Long-range interactions;   

DOI  :  10.1016/j.spa.2016.03.005

来源: Elsevier

【 摘 要 】

We investigate unbounded continuous spin-systems with infinite-range interactions. We develop a new technique for deducing decay of correlations from a uniform Poincare inequality based on a directional Poincare inequality, which we derive through an averaging procedure. We show that this decay of correlations is equivalent to the Dobrushin-Shlosman mixing condition. With this, we also state and provide a partial answer to a conjecture regarding the relationship between the relaxation rates of non-ferromagnetic and ferromagnetic systems. Finally, we show that for a symmetric, ferromagnetic system with zero boundary conditions, a weaker decay of correlations can be bootstrapped. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free