An Explicit Bound on the Logarithmic Sobolev Constant of Weakly Dependent Random Variables

We prove logarithmic Sobolev inequality for measures q(x) = dist(X) = exp ( −V (x) ) , x ∈ R, under the assumptions that: (i) the conditional distributions Qi(·|xj , j 6= i) = dist(Xi|Xj = xj , j 6= i) satisfy a logarithmic Sobolev inequality with a common constant ρ, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian V are not too large relative to ρ. 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 · ρ · (1 − δ). The logarithmic Sobolev constant of q can then be estimated from below by 1/2 ·ρ ·δ. This improves on earlier results by Th. Bodineau and B. Helffer, by giving an explicit bound, for the logarithmic Sobolev constant for q. Typeset by AMS-TEX 1 2 LOGARITHMIC SOBOLEV INEQUALITY