## 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.