An explicit bound on the Logarithmic Sobolev constant of weakly dependent random variables

Abstract

We prove logarithmic Sobolev inequality for measures $$q^n(x^n)=\text{dist}(X^n)=\exp (-V(x^n)),x^n\in {\mathbb{R}}^n,$$ under the assumptions that: (i) the conditional distributions $$Q_i(\cdot |x_j,j\ne i)=\text{dist}(X_i|X_j=x_j,j\ne i)$$ satisfy a logarithmic Sobolev inequality with a common constant $\rho$, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian $V$ are not too large relative to $\rho$. \bigskip Condition (ii) has the form that the norms of some matrices defined in terms of the mixed partial derivatives of $V$ do not exceed $1/2\cdotρ\cdot(1-\de)$. The logarithmic Sobolev constant of $q^n$ can then be estimated from below by $1/2\cdot \rho \cdot \delta$. This improves on earlier results by Th. Bodineau and B. Helffer, by giving an explicit bound, for the logarithmic Sobolev constant for $q^n$.

This paper has been withdrawn by the author because it was a preliminary version of arXiv:1206.4868

References in corpus (1)

• Measure concentration for Euclidean distance in the case of dependent random variables