97-646 Helffer B.

Remarks on decay of correlations and Witten Laplacians II\\ -- Analysis of the dependence on the interaction -- (44K, LATEX) Dec 24, 97

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Abstract. This is the continuation of a previous article \\ ``Remarks on decay of correlations and Witten Laplacians \\ -- Brascamp-Lieb inequalities and semiclassical limit --'' and devoted to the analysis of Laplace integrals attached to the measure $\exp - \Phi(X)\;dX$ for suitable families of phase $\Phi$ appearing naturally in the context of statistical mechanics. The main application treated in Part I was a semi-classical one ($\Phi=\Psi/h$ and $h\ar 0$) and the assumptions on the phase were related to weak non convexity.\\ We first analyze in the same spirit the case when the coefficient of the interaction $\Jg$ is possibly large and give rather explicit lower bounds for the lowest eigenvalue of the Witten Laplacian on $1$-forms. We also analyze the case $\Jg$ small by discussing first an unpublished proof of Bach-Jecko-Sj\"ostrand and then an alternative approach based on the analysis of a family of $1$-dimensional Witten Laplacians. We also compare with the results given by Sokal's approach. In part III of this serie, we shall analyze, in a less explicit way but in a more general context, applications to the logarithmic Sobolev inequality.

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