Renormalisation of random hierarchial systems
<p>This thesis considers a number of problems which are related to the study of random fractals. We define a class of iterations (which we call <em>random hierarchical systems</em>) of probability distributions, which are defined by applying a random map to a set of <em>k<...
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| Format: | Thesis |
| Language: | English |
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2003
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| _version_ | 1797064042321281024 |
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| author | Jordan, J Jordan, Jonathan |
| author_facet | Jordan, J Jordan, Jonathan |
| author_sort | Jordan, J |
| collection | OXFORD |
| description | <p>This thesis considers a number of problems which are related to the study of random fractals. We define a class of iterations (which we call <em>random hierarchical systems</em>) of probability distributions, which are defined by applying a random map to a set of <em>k</em> independent and identically distributed random variables. Classical examples of this sort of iteration include the Strong Law of Large Numbers, Galton-Watson branching processes, and the construction of random self-similar sets.</p> <p>In Chapter 2, we consider random hierarchical systems on ℝ, under the condition that the random map is bounded above by a random weighted mean, and that the initial distribution is bounded below. Under moment conditions on the initial distribution we show that there exists almost sure convergence to a constant.</p> <p>In Chapters 3 to 5 we consider the asymptotics of some examples of random hierarchical systems, some of which arise when considering certain properties of random fractal graphs. In one example, which is related to first-passage percolation on a random hierarchical lattice, we show the existence of a family of non-degenerate fixed points and show that the sequence of distributions will converge to one of these. The results of some simulations are reported in Chapter 7.</p> <p>Part III investigates the spectral properties of random fractal graphs. In Chapter 8 we look at one example in detail, showing that there exist localised eigenfunctions which lead to certain eigenvalues having very high multiplicity. We also investigate the behaviour of the Cheeger constants of this example. We then consider eigenvalues of homogeneous random fractal graphs, which preserve some of the symmetry of deterministic fractals. We then use relationships between homogeneous graphs and more general random fractal graphs to obtain results on the eigenvalues of the latter. Finally, we consider a few further examples.</p> |
| first_indexed | 2024-03-06T21:08:38Z |
| format | Thesis |
| id | oxford-uuid:3d5d122e-6c32-48c5-ae80-68c095a7131b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:08:38Z |
| publishDate | 2003 |
| record_format | dspace |
| spelling | oxford-uuid:3d5d122e-6c32-48c5-ae80-68c095a7131b2022-03-26T14:18:53ZRenormalisation of random hierarchial systemsThesishttp://purl.org/coar/resource_type/c_db06uuid:3d5d122e-6c32-48c5-ae80-68c095a7131bResearchFractalsEigenvaluesEnglishPolonsky Theses Digitisation Project2003Jordan, JJordan, Jonathan<p>This thesis considers a number of problems which are related to the study of random fractals. We define a class of iterations (which we call <em>random hierarchical systems</em>) of probability distributions, which are defined by applying a random map to a set of <em>k</em> independent and identically distributed random variables. Classical examples of this sort of iteration include the Strong Law of Large Numbers, Galton-Watson branching processes, and the construction of random self-similar sets.</p> <p>In Chapter 2, we consider random hierarchical systems on ℝ, under the condition that the random map is bounded above by a random weighted mean, and that the initial distribution is bounded below. Under moment conditions on the initial distribution we show that there exists almost sure convergence to a constant.</p> <p>In Chapters 3 to 5 we consider the asymptotics of some examples of random hierarchical systems, some of which arise when considering certain properties of random fractal graphs. In one example, which is related to first-passage percolation on a random hierarchical lattice, we show the existence of a family of non-degenerate fixed points and show that the sequence of distributions will converge to one of these. The results of some simulations are reported in Chapter 7.</p> <p>Part III investigates the spectral properties of random fractal graphs. In Chapter 8 we look at one example in detail, showing that there exist localised eigenfunctions which lead to certain eigenvalues having very high multiplicity. We also investigate the behaviour of the Cheeger constants of this example. We then consider eigenvalues of homogeneous random fractal graphs, which preserve some of the symmetry of deterministic fractals. We then use relationships between homogeneous graphs and more general random fractal graphs to obtain results on the eigenvalues of the latter. Finally, we consider a few further examples.</p> |
| spellingShingle | Research Fractals Eigenvalues Jordan, J Jordan, Jonathan Renormalisation of random hierarchial systems |
| title | Renormalisation of random hierarchial systems |
| title_full | Renormalisation of random hierarchial systems |
| title_fullStr | Renormalisation of random hierarchial systems |
| title_full_unstemmed | Renormalisation of random hierarchial systems |
| title_short | Renormalisation of random hierarchial systems |
| title_sort | renormalisation of random hierarchial systems |
| topic | Research Fractals Eigenvalues |
| work_keys_str_mv | AT jordanj renormalisationofrandomhierarchialsystems AT jordanjonathan renormalisationofrandomhierarchialsystems |