On the Scaling Limits of Random Planar Trees and Maps

Abstract

In this paper, we discuss connections between discrete and continuous random trees and how we can use that relationship to derive results about the asymptotics of a uniformly distributed random surface. Using Galton-Watson trees as our model for random discrete trees, we show their contour functions converge to a Brownian excursion. We then introduce real trees as a model for continuous trees and show that discrete trees converge in the Gromov-Hausdorff sense to a continuum real tree. Next, we discuss how putting label functions on discrete trees creates a bijection between random labeled trees and random planar maps, which can viewed as a discrete random surface. Finally, we use the scaling limit of random trees and this bijection to describe a uniformly distributed random surface called a Brownian map and present some properties about it