In an earlier work [J. Stat. Phys. 74, 349-409 (1994)], B. Helffer and the author studied correlations associated with the measure on , where is a smooth convex function, with special attention to the limit as tends to infinity. We observed (under suitable assumptions) that the expectation of a function can be obtained by solving the equation
with growing not too fast near infinity. In the present paper, we use more -methods and avoid excessive use of the maximum principle. A new observation (at least for the present author) is that the operator in (1) is a conjugated version of a Witten Laplacian in degree 0 and that the differentiated versions of (1) at least in some cases involve Witten Laplacians of higher degree .
A very natural idea is that in order to get more precise asymptotic results on the correlations, one must analyze the spectrum of the Witten Laplacian in degree 0 a little above the first spectral gap, and the main achievement of the present paper is in making a step in that direction. The main result of the paper describes the asymptotics of the exponential decay of the correlations between and as .