 97646 Helffer B.
 Remarks on decay of correlations and Witten Laplacians II\\
 Analysis of the dependence on the interaction 
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Dec 24, 97

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Abstract. This is the continuation of a previous article \\
``Remarks on decay of correlations and Witten Laplacians \\
 BrascampLieb 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
semiclassical 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unpublished proof of BachJeckoSj\"ostrand and then an alternative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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