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Title: On the Scaling Limits of Random Planar Trees and Maps
Authors: Lin, Ricky
Advisors: Sly, Allan
Department: Mathematics
Certificate Program: 
Class Year: 2023
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
Type of Material: Princeton University Senior Theses
Language: en
Appears in Collections:Mathematics, 1934-2023

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