MARK KAC SEMINAR

The combinatorial approach is a mathematically rigorous approach to study the two-dimensional Ising model in zero external field. Its origins can be traced back to a paper by Kac and Ward from 1952, but surprisingly, the method has never received much attention from the mathematics community. Yet, using the combinatorial approach, some classical results for the Ising model can be derived in a way which is more accessible and understandable for probabilists than the Onsager–Kaufman method. These results include Onsager's formula and the identification of the phase transition in terms of the behaviour of spin-spin correlations. In this lecture, I will explain the combinatorial identity that lies at the heart of the combinatorial approach, and discuss how some classical results can be derived using this identity.

The dynamics of disordered spin systems, in particular some mean field spin glasses, exhibits a phenomenon called ageing. It has been observed that a large class of such systems are effectively described by a simple model, the so-called REM-like trap model. I will explain that this is rather easily understood from the universality properties discussed in Lecture 1.