Random Surfaces and Quantum Loewner Evolution

Jason Miller (MIT)

Abstract :

What is the canonical way to choose a random, discrete, two-dimensional manifold which is homeomorphic to the sphere? One procedure for doing so is to choose uniformly among the set of surfaces which can be generated by gluing together $n$ Euclidean squares along their boundary segments. This is an example of what is called a random planar map and is a model of what is known as pure discrete quantum gravity. The asymptotic behavior of these discrete, random surfaces has been the focus of a large body of literature in both probability and combinatorics. This has culminated with the recent works of Le Gall and Miermont which prove that the $n \to \infty$ distributional limit of these surfaces exists with respect to the Gromov-Hausdorff metric after appropriate rescaling. The limiting random metric space is called the Brownian map.

Another canonical way to choose a random, two-dimensional manifold is what is known as Liouville quantum gravity (LQG). This is a theory of continuum quantum gravity introduced by Polyakov to model the time-space trajectory of a string. Its metric when parameterized by isothermal coordinates is formally described by $e^{\gamma h} (dx^2 + dy^2)$ where $h$ is an instance of the continuum Gaussian free field, the standard Gaussian with respect to the Dirichlet inner product. Although $h$ is not a function, Duplantier and Sheffield succeeded in constructing LQG rigorously as a random area measure. LQG for $\gamma=\sqrt{8/3}$ is conjecturally equivalent to the Brownian map and to the limits of other discrete theories of quantum gravity for other values of $\gamma$.

In this talk, I will describe a new family of growth processes called quantum Loewner evolution (QLE) which we propose using to endow LQG with a distance function which is isometric to the Brownian map. I will also explain how QLE is related to DLA, the dielectric breakdown model, and SLE.

Based on joint works with Scott Sheffield.