Helffer B.
Remarks on  decay of correlations and Witten Laplacians III\\
--Applications to logarithmic Sobolev inequalities-- 
(40K, LATEX)

ABSTRACT.  This is the continuation of our two previous articles devoted to the use of
Witten Laplacians for analyzing Laplace integrals in statistical
mechanics. 
The main application treated in Part I was a semi-classical one. 
The second application was more perturbative in spirit and gave very
 explicit estimates for the lower bound of the Witten Laplacian in the
case of a quartic model. 
We shall relate in this third part our studies of the Witten Laplacian
 with the existence of uniform logarithmic Sobolev inequalities
through a criterion of B. Zegarlinski. 
More precisely, our main contribution is to  show how to control the
decay of correlations uniformly  with respect to various parameters,
 under a natural condition of strict convexity at $\infty$ of the
single-spin phase and when the nearest neighbor interaction is small 
enough.