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 ...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.展开更多
This paper deals with the optimal exponential convergence rate βto the equilibrium state in Boltzmann-Shannon entropy for general Markov semigroups. We prove a variational formula of β, and then discuss the relation...This paper deals with the optimal exponential convergence rate βto the equilibrium state in Boltzmann-Shannon entropy for general Markov semigroups. We prove a variational formula of β, and then discuss the relation among β,spectral gap λ and logarithmic Sobolev constant α, which is read as λ≥β≥α.展开更多
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
文摘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.
基金This work was supported in part by the National Natural Science Foundation ofChina (Grant Nos. 19771008, 19971025) the Research Fund for the Doctoral Program of Higher Education (Grant No. 96002704) Fok Ying-Tung Youth Foundation.
文摘This paper deals with the optimal exponential convergence rate βto the equilibrium state in Boltzmann-Shannon entropy for general Markov semigroups. We prove a variational formula of β, and then discuss the relation among β,spectral gap λ and logarithmic Sobolev constant α, which is read as λ≥β≥α.
