Abstract

In this minicourse, we introduce and relate two closely connected models, the nonlinear supersymmetric hyperbolic sigma model on the one hand and the vertex-reinforced jump process on the other hand. To prepare the study of the first model, an introduction to supersymmetry will also be given, which deals with both, commuting and anticommuting variables with values in a Grassmann algebra, and first order linear differential operators mixing them. The first model is then a statistical mechanics type spin model with spin components taking values in this Grassmann algebra. The second model is a stochastic process for a particle with memory jumping in continuous time on agraph. The memory is encoded in the local times the particle has already spent at the vertices of the graph; frequently visited sites have a larger rate to be visited again.  Asymptotic ratios of local times of the second model have a direct interpretation in terms of spin variables of the first model in appropriate coordinates. We explain some bounds in the first model and show how they can be used to understand the recurrence and transience behaviour in the second model. The minicourse includes some joint work with Margherita Disertori and Pierre Tarrès.