Remarks on decay of correlations and Witten Laplacians Brascamp-Lieb inequalities and semiclassical limit
type: Item
instance of: scholarly article
title: Remarks on decay of correlations and Witten Laplacians Brascamp-Lieb inequalities and semiclassical limit
zbMATH Open document ID: 0921.35141
author: Bernard Helffer
published in: Journal of Functional Analysis
publication date: 1999-09-29T00:00:00Z
review text: The paper under review gives an analysis about the integral/expectation with respect to the measure \(d\mu(x)= e^{-\Phi(x)}dx/ \int e^{-\Phi(x)}dx\) on \(\mathbb{R}^m\), where \(dx\) is the Lebesgue measure and \(\Phi(x)\) is a real-valued \(C^2\) function on \(\mathbb{R}^m\). It corresponds to a ``partition function'' stemming from statistical mechanical lattice models. Put \(\langle g\rangle= \int g(x)d \mu(x)\) and define the covariance and variance by \(\text{cov}(f,g)= \langle(f-\langle f\rangle)(\overline g-\langle \overline g\rangle) \rangle\) and \(\text{var} g=\text{cov} (g,g)\). Then in this kind of problem an important role is played by the Brascamp-Lieb (BL) inequality: \[ \text{var} g \leq\biggl \langle \nabla g \cdot\bigl((\text{Hess} \Phi)^{-1}\nabla \overline g\bigr) \biggr\rangle \] for \(g\in C^1(\mathbb{R}^m)\) with \(\text{var} g<\infty\), which holds when \(\Phi(x)\) is strictly convex. However, the author has in mind as \(\Phi(x)\) (though with the measure \(d\mu(x)= e^{-\Phi(x)/h} dx/\int e^{-\Phi (x)/h}dx\) instead for a sufficiently small parameter \(h>0)\) the following function \[ \Phi(x)= \sum_{j\in \Lambda} (\lambda_jx_j^4+ \nu_jx_j^2) +{J\over 2} \sum_{j\sim k} (x_j-x_{j+1})^2 \] on \(\mathbb{R}^\Lambda\) with a cube \(\Lambda\) of \(\mathbb{Z}^d\), \(|\Lambda |=m\), with periodic boundary condition, where \(J>0\) and \(\lambda_j\geq\underline\lambda\), \(\nu_j\geq-\varepsilon_0h\) uniformly in \(j\) for a constant \(\underline\lambda>0\) and some sufficiently small \(\varepsilon_0 >0\). The second sum \(\sum_{j\sim k}\) is taken over the nearest neighbours \(j\) and \(k\) in \(\Lambda\). This \(\Phi(x)\) is not convex in general. He points out, by a sketchy argument based on an idea of \textit{J. Sjöstrand} [St. Petersbg. Math. J. 8, No. 1, 123-147 (1997; Zbl 0877.35084) and Algebra Anal. 8, No. 1, 160-191 (1996)] and the author [Recent results and open problems on Schrödinger operators, Laplace integrals, and transfer operators in lar…
reviewed by: Takashi Ichinose
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 26D07
zbMATH DE Number: 1199822
zbMATH Keywords: Brascamp-Lieb inequality
zbMATH Keywords: lattice model
zbMATH Keywords: correlation decay
zbMATH Keywords: Witten Laplacian
MaRDI profile type: Publication
full work available at URL: jfan.1997.3239
OpenAlex ID: W2091425527
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DOI: 10.1006/JFAN.1997.3239
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description: scientific article; zbMATH DE number 1199822