Spectral gap and logarithmic Sobolev constant for continuous spin systems

TL;DR

This paper investigates the spectral gap and logarithmic Sobolev constant for continuous spin systems, providing general estimates, exact dimension one results, and explicit bounds for specific infinite-dimensional models.

Contribution

It introduces a general method to estimate spectral gaps from one-dimensional marginals and offers precise results for key models, advancing understanding of infinite-dimensional spin systems.

Findings

A simple estimate for spectral gap based on one-dimensional marginals

Exact order of spectral gap in one dimension established

Explicit bounds for spectral gap and Sobolev constant in two models

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.