Ben Lees (School of Mathematics, University of Leeds) – Discrete lattice models as complex spin systems

It has long been known that some important spin systems on lattices, such as the spin-O(N) model, have representations in terms of discrete structures (loops, paths etc) on the lattice. This has been a fruitful connection, however many discrete models on lattices, such as percolation, the dimer model,  spatial random permutations, and others, are also of considerable interest. In these cases there was no precise link to spin systems, either because the motivation came from elsewhere, or because these are toy models that have taken on a life of their own. In this talk I will present a spin system where spins take continuous complex values. By choosing the measures appropriately, virtually any discrete system of interest can be obtained as a special case. These spin systems then lend themselves naturally to several of the well developed tools of classical spin systems. This is joint work with Volker Betz.