Scaling limits in random conformal geometry

The goal of this project is to explore geometric properties of certain random structures in two dimensions. These structures are ultimately of interest as models in theoretical physics, but are motivated at least as much by the possibility that they are fundamental, universal mathematical objects. The crucial feature of these structures is their expected conformal invariance: that is, invariance under deformations of the plane that locally preserve angles. A proof of this property usually implies a description of their scaling limit which simultaneously unlocks their key properties, via the theory of Schramm--Loewner Evolution. This project will explore how certain random systems acquire conformal invariance in the large scale limit, and how the inherent randomness of these systems interacts with this conformal geometry. We will build on some remarkable developments which have taken place in recent years in the field, in particular in connection with Liouville quantum gravity and more generally the study of conformally invariant random processes in the plane - especially the Gaussian free field, a continuum random generalised function that can be defined in any given domain of the plane. We will focus on two interrelated aspects of this question: on the one hand, Liouville Brownian motion, a canonical notion of diffusion in Liouville quantum gravity; and on the other hand, the dimer model, a classical model of statistical mechanics equivalent to a random tiling, whose limiting conformal structure is governed by the so-called Imaginary Geometry, relating Schramm--Loewner Evolution to the Gaussian free field.

The goal of this project is to explore geometric properties of certain random structures in two dimensions. These structures are ultimately of interest as models in theoretical physics, but are motivated at least as much by the possibility that they are fundamental, universal mathematical objects. The crucial feature of these structures is their expected conformal invariance: that is, invariance under deformations of the plane that locally preserve angles. A proof of this property usually implies a description of their scaling limit which simultaneously unlocks their key properties, via the theory of Schramm--Loewner Evolution. This project will explore how certain random systems acquire conformal invariance in the large scale limit, and how the inherent randomness of these systems interacts with this conformal geometry. We will build on some remarkable developments which have taken place in recent years in the field, in particular in connection with Liouville quantum gravity and more generally the study of conformally invariant random processes in the plane -- especially the Gaussian free field, a continuum random generalised function that can be defined in any given domain of the plane. We will focus on two interrelated aspects of this question: on the one hand, Liouville Brownian motion, a canonical notion of diffusion in Liouville quantum gravity; and on the other hand, the dimer model, a classical model of statistical mechanics equivalent to a random tiling, whose limiting conformal structure is governed by the so-called Imaginary Geometry, relating Schramm--Loewner Evolution to the Gaussian free field.