REMARKS ON DECAY OF CORRELATIONS AND WITTEN LAPLACIANS II — ANALYSIS OF THE DEPENDENCE ON THE INTERACTION

Abstract

This is the continuation of previous notes on the subject referred as [10] and devoted to the analysis of Laplace integrals attached to the measure exp -Φ(X)dX for suitable families of phase Φ appearing naturally in the context of statistical mechanics. The main application treated in [10] was a semi-classical one (Φ=Ψ/h and h→0), and the assumptions on the phase were related to weak non-convexity. We analyze here in the same spirit the case when the coefficient of the interaction [Formula: see text] possibly is large and give rather explicit lower bounds for the lowest eigenvalue of the Witten Laplacian on 1-forms. We also analyze the case small [Formula: see text] by discussing first an unpublished proof of [2] and then an alternative approach based on the analysis of a family of 1-dimensional Witten Laplacians. We also compare our results to those by Sokal's approach. In the last paper of this series [11], we shall analyze, in a less explicit way, but in a more general context, applications to the logarithmic Sobolev inequality.