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Random walks on decorated Galton-Watson trees
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Archer, Eleanor (2020) Random walks on decorated Galton-Watson trees. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3492260~S15
Abstract
The purpose of this thesis is to study random walks on “decorated” Galton-Watson trees with critical offspring distribution in the domain of attraction of an α-stable law for some α ∈ (1, 2).
In Chapters 2 and 3 we give some background on the topologies and models used in the thesis. In Chapter 4 we consider a specific example: stable looptrees. We prove a scaling limit result for convergence of random walks on discrete looptrees to convergence of Brownian motion on continuum looptrees. We then construct a detailed investigation of the limiting Brownian motion, in particular obtaining detailed bounds on the transition density and on the spectrum of the associated Laplacian. Along the way, we also prove precise volume asymptotics.
In Chapter 5 we construct the local limit of compact stable looptrees which we call infinite stable looptrees. In particular, this allows us to show that the operations of taking scaling limits and local limits of discrete and continuum looptrees can be done in either order or in combination. As a result, we are also able to prove similar limit results for stochastic processes on these spaces. Moreover, we are able to apply the local limit result to obtain limiting results for the volume of a small ball and the small-time on-diagonal transition density for compact stable looptrees.
In Chapter 6 we consider the main general model of interest: that of a decorated Galton-Watson tree. In this chapter we formulate some assumptions regarding the graphs used for “decoration”, and then prove some results establishing the volume growth exponent, random walk displacement exponent and spectral dimension for these decorated Galton-Watson trees.
In Chapter 7 we give some brief comments on future research directions.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Random walks (Mathematics), Trees (Graph theory), Branching processes | ||||
Official Date: | June 2020 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Croydon, David A. | ||||
Format of File: | |||||
Extent: | vii, 218 leaves : illustrations (chiefly colour) | ||||
Language: | eng |
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