Correlation decay and gap of the transfer operator

The aim of this article is to present a new proof for a universal estimate of the gap between the two largest eigenvalues of the Kac operator in the case of a convex potential. This is done by means of a careful analysis of the technique of the transfer matrix in connection with the recent study of the decay of correlations associated to Laplace integrals by Helffer-Sjöstrand and Sjöstrand.