Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics

This paper is devoted to problems coming from statistical mechanics. The transfer matrix (or transfer operator) approach consists in reducing the analysis of asymptotic properties of statistical systems to the analysis of the spectral properties of their transfer operator. Sometimes the new problem appears to have a semi-classical nature. Although the problem is similar to the semiclassical study of Kac's operator presented in our paper with M. Brunaud [BruHe] which was devoted to the study of \(\) for h small, new features appear for the model \(\). Our first results concern semi-classical analysis of the ground state for this second operator. We then analyze the two models in the large dimension situation. One basic technique is Sjostrand's formalism of the 0-standard functions as introduced in [Sj5]. The one-dimensional case was presented in [He11].

We present here a review of two problems motivated by 2D models for high T, superconductivity. The first part concerns the energy spectrum of 2D Bloch electrons in a uniform magnetic field. A semiclassical analysis provides a qualitative as well as a quantitative understanding of this spectrum. In the second part we make the case for the application of “Quantum Chaos" to strongly correlated fermion systems. It is illustrated by the level spacing distribution for the t - J model in two dimensions.