Some estimates of the logarithmic Sobolev constants on manifolds with boundary and an application to the Ising models
type: Item
instance of: scholarly article
title: Some estimates of the logarithmic Sobolev constants on manifolds with boundary and an application to the Ising models
zbMATH Open document ID: 1006.58018
author: Yuzuru Inahama
published in: Journal of Mathematics of Kyoto University
publication date: 2003-03-09T00:00:00Z
review text: Let \(M\) be a compact \(d\)-dimensional Riemannian manifold with smooth boundary \(\partial M\), \(U\) a potential function on \(M\), and let \(dm^U\) be \(e^{-U}dm\). Then the logarithmic Sobolev inequality for \(a^U\) is \[ \int_Mf^2 \log \left({f^2\over \|f\|^2_{L^2 (m^U)}}\right) dm^U\leq{2\over \alpha} a^U(f,f), \] \[ a^U(f,g)= \int_M(\nabla f\mid\nabla g) dm^U. \] The logarithmic Sobolev constant \(\alpha(U)\) is the best constant in this inequality. In this paper, estimates of logarithmic Sobolev constants of \textit{J. D. Deuschel} and \textit{D. W. Stroock} [J. Funct. Anal. 92, No. 1, 30-48 (1990; Zbl 0705.60066)] for manifolds without boundary is generalized to manifolds with boundary by adding a certain term which includes the second fundamental form. Then considering the Gibbs measures on \(M^{\mathbb{Z}^\nu}\) determined by a finite range and shift-invariant potential, these results are applied for finite dimensional manifolds to the Ising model. The outline of the paper is as follows: In section 2, some estimates from below of the second fundamental form \(A(\theta,\eta)=\frac 12\nabla_N (\theta|\eta)\), \(( \theta\mid N)= (\eta\mid N)=0\), where \(n\) is the inner normal vector on \(\partial M\) by which the Neumann boundary condition is given, are prepared (Lemma 2.4). By using these estimates, estimates of spectral gap constants are given (Prop. 2.6). Here, the spectral gap constant \(C(U)\) is the best constant to hold the spectral gap inequality \[ \bigl\|f-\langle f \rangle_{ m^U} \bigr\|_{L^2 (m^U)}\leq{1\over C}a^U (f,f),\quad \nabla_Nf=0. \] In section 3, it is shown if \(f\) satisfies the Neumann boundary condition and is positive, then \[ \biggl\langle f\bigl\|\text{Hess}(\log f)\bigr \|^2 \biggr \rangle_m\geq {4\over d+2} \;2\bigl \|\text{Hess} f^{1/2} \bigr\|^2+ (\Delta f^{1/2})^2 \biggr\rangle_m \] (Lemma 3.2). By this inequality and Lemma 2.4, several versions of the logarithmic Sobolev inequalities are derived. Then by using the estima…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 1796034
zbMATH Keywords: logarithmic Sobolev constants
zbMATH Keywords: Ising model
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: spectral gap constants
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250517638
OpenAlex ID: W1574411128
DOI: 10.1215/KJM/1250517638
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description: scientific article; zbMATH DE number 1796034