A random walk through combinatorial probability
- This thesis comprises a collection of three stand-alone problems in combinatorial probability. A common aspect to all three problems is the use of perturbation based techniques to show that a quantity we wish to estimate is asymptotically equivalent to another known quantity. The first problem generalises Arratia, Barbour and Tavare's (2000) theory of logarithmic combinatorial structures, showing that conditional independence of component counts is not necessary for results such as the central limit theorem, the Poisson-Dirichlet limit and the Erdos-Turan limit, and unifying these theorems within a single universality result that sheds greater insight into why the same limit theorem should hold for all families of combinatorial structures in this class. The second problem studies the Hopfield model in the limit where the number of patterns is superlinear in the number of sites, computing the asymptotic behaviour of the free energy by applying Guerra and Toninelli's (2002) interpolation technique to the related Sherrington-Kirkpatrick model. The third problem describes a new strategy for sampling combinatorial structures, including many simple demonstrative examples, and with particular application to the case of labelled graphs with given degree sequence, where we describe an algorithm whose domain of applicability and running time compare quite favourably to existing algorithms in the literature.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Zhao, James Yuanjie
|Stanford University, Department of Mathematics.
|Statement of responsibility
|James Yuanjie Zhao.
|Submitted to the Department of Mathematics.
|Thesis (Ph.D.)--Stanford University, 2013.
- © 2013 by James Yuanjie Zhao
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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