In the last several decades, two canonical theories of random surfaces have emerged. The first, so-called Liouville quantum gravity (LQG), has its roots in conformal field theory and string theory from the 1980s and 1990s. The second, so-called random planar maps (RPM), has its roots in combinatorics from the 1960s. There has been immense progress in recent years in making rigorous sense of LQG, in the study of the large-scale behavior of RPM, and developing connections between them as well as with other mathematical objects such as the Schramm-Loewner evolution (SLE).
The purpose of this project is to study stochastic processes on LQG and RPM.
The first part of the proposed research is focused on developing a theory of growth processes on LQG, the so-called quantum Loewner evolution (QLE). QLE, introduced in joint work with Sheffield, is a family of processes which conjecturally describe the scaling limits of discrete growth processes such as diffusion limited aggregation (DLA), the Eden model, and the dielectric breakdown model (DBM) on LQG. QLE has proved to be a powerful tool in the study of LQG and RPM and was used in joint work with Sheffield to unite LQG with gamma=sqrt(8/3) with the Brownian map, the metric space scaling limit of random planar maps. Nevertheless, the development of QLE is still in its infancy and many important problems remain to be solved.
It has long been conjectured that large RPM equipped with a statistical physics model, such as percolation or a uniform spanning tree, embedded into the plane in a conformal manner should be described by a form of LQG decorated by an SLE. The embedding problem is intimately connected to understanding random walk on a RPM, which has proved to be challenging. The second part of the proposal is aimed at developing new methods to settle long-standing questions about random walk on RPM, and ultimately the embedding problem for RPM.
Fields of science
Call for proposal
See other projects for this call