UBC Theses and Dissertations
Hyperbolic random maps Ray, Gourab
Random planar maps have been an object of utmost interest over the last decade and half since the pioneering works of Benjamini and Schramm, Angel and Schramm and Chassaing and Schaeffer. These maps serve as models of random surfaces, the study of which is very important with motivations from physics, combinatorics and random geometry. Uniform infinite planar maps, introduced by Angel and Schramm, which are obtained as local limits of uniform finite maps embedded in the sphere, serve as a very important discrete model of infinite random surfaces. Recently, there has been growing interest to create and understand hyperbolic versions of such uniform infinite maps and several conjectures and proposed models have been around for some time. In this thesis, we mainly address these questions from several viewpoints and gather evidence of their existence and nature. The thesis can be broadly divided into two parts. The first part is concerned with half planar maps (maps embedded in the upper half plane) which enjoy a certain domain Markov property. This is reminiscent of that of the SLE curves. Chapters 2 and 3 are mainly concerned with classi cation of such maps and their study, with a special focus on triangulations. The second part concerns investigating unicellular maps or maps with one face embedded in a high genus surface. Unicellular maps are generalizations of trees in higher genera. The main motivation is that investigating such maps will shed some light into understanding the local limit of general maps via some well-known bijective techniques. We obtain certain information about the large scale geometry of such maps in Chapter 4 and about the local limit of such maps in Chapter 5.
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