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

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 $$Q_i(\cdot \mathrm{\mid }{x}_j,j\mathrm{\ne }i)=\text{dist}(X_i\mathrm{\mid }{X}_j=x_j,j\mathrm{\ne }i)$$ 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 $1\mathrm{/}2\cdot \rho \cdot (1-\text{de})$. The logarithmic Sobolev constant of $q^n$ can then be estimated from below by $1\mathrm{/}2\cdot \rho \cdot \delta$. This improves on earlier results by Th. Bodineau and B. Helffer, by giving an explicit bound, for the logarithmic Sobolev constant for $q^n$.