Spectral Gap and Logarithmic Sobolev Constant for Continuous Spin Systems  

作　　者：Mu Fa CHEN 

机构地区：[1]School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2008年第5期705-736,共32页数学学报（英文版）

基　　金：the Creative Research Group Fund of the National Natural Science Foundation of China (No.10121101);the"985"Project from the Ministry of Education of China

摘　　要：The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap＇of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of this topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one （Proposition 1.4）, and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented （Theorems 6.2 and 6.3）. Since we are interested in explicit estimates, the computations become quite involved. A long section （Section 4） is devoted to the study of the spectral gap in dimension one.The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap＇of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of this topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one （Proposition 1.4）, and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented （Theorems 6.2 and 6.3）. Since we are interested in explicit estimates, the computations become quite involved. A long section （Section 4） is devoted to the study of the spectral gap in dimension one.

关 键 词：spectral gap logarithmic Sobolev constant spin system principal eigenvalue 

分 类 号：O17[理学—数学]