Abstract

Many models in statistical physics, such is the Ising model or percolation, are defined on graphs. The graphs are usually taken to be deterministic, regular lattices, but it is also possible to define the same models on random graphs. We often want to study properties of the scaling limits of these models – the limits where the lattice size is taken to zero.

When the graph used is a regular lattice, the geometry of the scaling limit is Euclidean. However, when we use a random graph, the geometry of the scaling limit is conjectured to be that of “Liouville quantum gravity,” which we would like to view as a random Riemann surface.

How to view this surface even as a metric space is still an open question but, due to work by Duplantier & Sheffield and Rhodes & Vargas, we have an area measure for the surface. Despite the fact that we only have an area metric for this surface, it is still possible to construct a Brownian motion on it, as shown by Berestycki and Garban, Rhodes & Vargas.

I will give a brief overview of the construction of the area measure from the Gaussian free field, and some of its properties, and also the construction of the Liouville Brownian motion and the specific aspect of it that I am studying.