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 We also analyze the case \$\Jg\$ small by discussing first an unpublished proof of Bach-Jecko-Sj\"ostrand and then an alternative approach based on the analysis of a family of \$1\$-dimensional Witten Laplacians.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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