Spectral gap and logarithmic Sobolev constant for continuous spin systems

Abstract: 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 the 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.