Random Surfaces, the Loewner Equation, and Embedded Trees

Vivian Healey (Brown University)

Abstract :

The question of how to construct a random Riemann surface has given rise to the study of random maps--random graphs embedded in surfaces. In particular, physicists use these discretizations of surfaces to model discrete quantum gravity. Using well-studied bijections, like the one due to Schaeffer, can reduce the study of certain random planar maps to the study of plane trees. However, when studying the limiting objects of trees (the continuum random tree) and planar maps (the Brownian map) the limits are those of metric spaces, and the specific geometries of the original combinatorial constructions are obscured.

On the other hand, the theory of the Loewner differential equation and Schramm-Loewner evolution (SLE) offers an extensive description of random paths in the upper half-plane. In this talk I will use Loewner theory to describe embeddings of trees in the upper half-plane. In particular, I will offer a candidate for a canonical isometric embedding of a class of discrete trees in the upper half-plane and discuss progress toward finding their limit, an isometric embedding of the continuum random tree.