Remarks on decay of correlations and Witten Laplacians Brascamp-Lieb inequalities and semiclassical limit
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 large dimensions. Demuth, Michael (ed.) et al., Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras. Berlin: Akademie Verlag. Math. Top. 11, 11-162 (1996)], that one can derive (BL) with a better upper bound than that on the right-hand side, even when \(\Phi(x)\) is weakly convex and in a certain case, weakly non-convex. It lies in the expression \[ \text{var} g= \int A_1^{-1} \nabla g\cdot \nabla\overline ge^{-\Phi(x)} dx/\int e^{-\Phi(x)} dx, \] which is to be verified, where \(A_1=\nabla \Phi\cdot \nabla-\Delta +\text{Hess} \Phi\) is a Witten Laplacian on one-forms. It turns out that \(A_1\) can be realized as a strictly positive selfadjoint operator. Hence (BL) follows not only in the strictly convex case, because \(A_1\geq\text{Hess} \Phi\geq \rho_0\) with constant \(\rho_0>0\) being the minimum of the eigenvalues of the matrix \(\text{Hess} \Phi(x)\), but also in the general case, because the lowest eigenvalue \(\rho_1\) of \(A_1\) is positive. In fact, the author shows that this argument applies to the above \(\Phi(x)\), concluding, for instance, that \(\text{var} g\leq C/| \Lambda|\) for \(g(x)={1\over |\Lambda |}\sum_{j \in \Lambda}x_j\) and also that \(|\text{cov}(x_1,x_j) |\leq De^{-jh/D}\) for every pair \((j,m)\) with \(1\leq j\leq m/2\), where \(\Lambda=\{1, \dots,m\}\).
scientific article; zbMATH DE number 1199822
scientific article