Around a Stationary Phase Theorem in Large Dimension

In this article, we want to present a stationary phase theorem appearing in a recent article by J. Sjِstrand (Ann. Inst. H. Poincaré Phys. Théor., in press) and explain the role it can play in the context of statistical mechanics. This theorem is also technically connected to the semi-classical determination of eigenvalues for a Schrِdinger operator in large dimension and also, in the same spirit, with small or large temperature expansions.