P03

## Emergence of macroscopic phenomena in the non-linear hyperbolic
supersymmetric sigma model

### Description Phase 2

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 the first phase, we considered a hierarchical model, where the
renormalisation operation becomes explicitly controllable thanks to the underlying
supersymmetries. This approach allowed us already to prove some
results on the vertex-reinforced jump process and the non-linear supersymmetric
hyperbolic sigma model not only at the hierarchical level, but also on
the euclidean lattice with long-range jumps. In the second phase, we plan to
identify the critical point in the hierarchical model, extend existing results
from locally finite graphs to the long-range case, study the recurrent regime
of vertex-reinforced jump processes with long-range jumps and analyse
which properties of the hierarchical lattice are universal and survive in the
Euclidean lattice with finite-range interaction.
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.

### Description Phase 1

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.

### Preprints/Publications

Margherita Disertori, Franz Merkl, Silke W.W. Rolles:
Transience of vertex-reinforced jump processes with long-range jumps
(05/2023)

Margherita Disertori, Franz Merkl, Silke W.W. Rolles:
The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions
(11/2021)
published