Spins, fields and loops

The goal of this project is the mathematical study of spin models, random surfaces and loops, and their relationship. Think of a random collection of directions assigned to each point of a square lattice. The directions (or arrows) represent magnetic moments of particles, and their behaviour changes when one increases the temperature of the physical system. It was famously predicted (which lead to the Nobel prize in physics in 2016) that this change is caused by a complicated interaction between vortices and antivortices (points on the lattice where the random directions makes a full turn in a small area). This is now called a topological phase transition. A fascinating mathematical construction called duality relates this behaviour to the study of randomly fluctuating surfaces. Through that link we plan address several open questions about the large scale behaviour of both the spin model and the random surface. When the lattice is cubic and not square, it is known that the phase transition is of a completely different nature. At low temperature, the arrows spontaneously organise and point in the same direction. An important question in the field is to understand the arrows exactly at the temperature of the transition. It is expected that the arrows point in all directions and look disordered. We plan to address this question using a newly discovered representation of the model as a collection of randomly fluctuation loops.