Lattice spin models

Background: My research is at the interface of probability theory and physics of many-particle systems (also called "statistical mechanics"). The latter are characterised by many degrees of freedom and an intrinsic uncertainty about the system's microscopic state, which is mathematically modelled by probability distributions. A natural example is a gas of atoms, perhaps interacting by attractive and repulsive forces. While it is essentially impossible to describe the position of every single atom (due to their sheer numbers), one may still ask interesting questions about the macroscopic behaviour of the system. In particular, one might wish to understand how the phase transition between a solid and a liquid/gaseous state at some temperature can be explained by the microscopic model.
Instead of atomic gases, my work focuses more on so-called "lattice spin-systems". These models are used to study the magnetic behaviour of materials such as iron, but also have more abstract connections to quantum field theory and particle physics. Similar to models of atomic gases, these systems describe a large number of interacting particles and often exhibit macroscopic phase transitions. In our research, we are trying to rigorously establish results about the phase transitions of these models. This includes statements about the existence of such a transition, the "critical" behaviour close to the transition, the high/low temperature behaviour, et cetera.

Approach: The study of lattice spin models has been active for several decades, hence there is already a large toolbox available. The methods span from the mathematical discipline of analysis to probability theory to statistics and more. We are combining many of these known methods with modern insights from the field of percolation theory (i.e. the study of random geometric structures) and supersymmetry in order to tackle previously poorly understood models.

Novel Content: In particular the usage of supersymmetry in a probabilistic context is a quite modern and not yet fully acknowledged idea. We would like to continue applying this method and show how it can be used in already well-known models from physics in order to obtain new results.