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

Katalin Marton

Published 2006-05-15, updated 2015-06-22Version 2

We prove logarithmic Sobolev inequality for measures $$ q^n(x^n)=\text{dist}(X^n)=\exp\bigl(-V(x^n)\bigr), \quad x^n\in \Bbb R^n, $$ under the assumptions that: (i) the conditional distributions $$ Q_i(\cdot| x_j, j\neq i)=\text{dist}(X_i| X_j= x_j, j\neq i) $$ 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 $1/2\cdot\rho\cdot(1-\de)$. 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 $q^n$.