Abstract

The classical random walk isomorphism theorems are a collection of bizarre distributional identities which relate observables of a simple random walk + its local time field to associated observables of a Gaussian Free Field, a spin system taking values in Euclidean space. In this talk, I will present a new and very simple framework for constructing these isomorphism theorems, where they are realised in terms of the continuous symmetries of the GFF . The key advantage of this framework is that it does not rely on explicit Euclidean/Gaussian computations, and as a result, allows us to extend the classical results to hyperbolic and spherical geometries. Here, the corresponding random walks are no longer Markovian: they are the vertex-reinforced and vertex-diminished jump processes. I will also discuss the supersymmetric versions of these spin systems, and present some simple applications of the results.