Mathematisches Kolloquium

Zusammenfassung: Some thirty years ago, a natural Laplacian operator has become a central object of the mathematical study of classical statistical mechanics (as part of probability). The variant chosen by Helffer and Sjöstrand in the early 1990ies is the Witten Laplacian found in a seminal paper of Witten a decade before in a geometric context. The link Helffer and Sjöstrand established, known as the Helffer-Sjöstrand Formula, is an identity between (truncated) correlation functions and the resolvent of the Witten Laplacian. Through this identity, spectral theory of linear operators becomes a central mathematical tool for the analysis of classical mechanical systems.
In the lecture, the link between classical statistical mechanics and spectral theory is reviewed on the example of an Ising-type model with real-valued spins. Basic questions such as existence of thermodynamic limits and thermodynamic potentials and the presence or absence of phase transistions are addressed. Advantages and disadvantages of the method and its power will be discussed.
New results presented here are joint work with Fiona Gottschalk.

Everyone is invited! Join us for coffee and tea in the common room (B448) half an hour before the talk.