Emergence of macroscopic phenomena in the non-linear hyperbolic
supersymmetric sigma model
The non-linear supersymmetric hyperbolic sigma model introduced by Zirnbauer in 1991 can be seen as a statistical mechanical model of spins taking values on a supersymmetric non-linear manifold. This model has deep connections to a class of stochastic processes with memory called vertex-reinforced jump processes. These connections were successfully used to prove a phase transition in dimension larger or equal than 3 for the vertex-reinforced jump process. Nevertheless many important questions still remain open.
A powerful but difficult tool to study statistical mechanics models is the renormalization group. Our plan is to modify this approach in such a way that it applies to the non-linear supersymmetric hyperbolic sigma model. We expect this to help understanding the behavior of the model near the phase transition. In a first step, we will consider a simplified problem on a hierarchical lattice, where the construction is expected to be more accessible.
We have been working intensively, together and separately, on the non-linear supersymmetric hyperbolic sigma model and the connected vertex-reinforced jump process. One of us has also publications on the renormalization group in a different context.