Thesis Open Access

Large Deviations of Convex Hulls of Random Walks and Other Stochastic Models

Schawe, Hendrik


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        <foaf:name>Schawe, Hendrik</foaf:name>
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    <dct:title>Large Deviations of Convex Hulls of Random Walks and Other Stochastic Models</dct:title>
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        <foaf:name>Hartmann, Alexander K.</foaf:name>
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        <foaf:name>Krug, Joachim</foaf:name>
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    <dct:description>&lt;p&gt;In the thesis at hand Monte Carlo methods originating from statistical&lt;br&gt; physics are applied to study various problems in far more detail than&lt;br&gt; before. While all those problems have in common that they were up to&lt;br&gt; now mainly studied in regards to the mean values of some observable, in&lt;br&gt; this thesis the full distribution including very rare events with&lt;br&gt; probabilities in the order of 10&lt;sup&gt;-100&lt;/sup&gt; and smaller are obtained and&lt;br&gt; discussed.&lt;/p&gt; &lt;p&gt;The first and largest project of this thesis is about the distribution&lt;br&gt; of the volume and surface of the convex hulls around the traces of&lt;br&gt; random walks. The first part of this project looks at the hulls of&lt;br&gt; standard random walks. For this rather simple model much progress was&lt;br&gt; made in the last decades and it is the only problem of this thesis for&lt;br&gt; which prior numerical results of the whole distribution exists in the special&lt;br&gt; case of two dimensions. Therefore this thesis focuses on a generalization&lt;br&gt; to higher dimensions. The second part of this project scrutinizes more&lt;br&gt; complicated types of random walks which interact with their past&lt;br&gt; trajectory. This interaction makes these random walks suitable as models&lt;br&gt; for, e.g., polymers. The same interaction also leads to an increased&lt;br&gt; difficulty in obtaining results analytically, such that the numerical&lt;br&gt; examination of the whole distribution seems worthwhile.&lt;/p&gt; &lt;p&gt;The second project examines the distribution of the ground-state energy&lt;br&gt; of a generalized random-energy model, a toy model from statistical&lt;br&gt; physics with applications to phase transitions and spin glasses.&lt;br&gt; There we find a universal asymptotic form for the distribution of the&lt;br&gt; ground-state energies in the limit of large systems, only dependent via&lt;br&gt; two parameters on the behavior of the underlying distribution of&lt;br&gt; the single energy levels in the system.&lt;/p&gt; &lt;p&gt;The third project scrutinizes the distribution of the length of the&lt;br&gt; longest increasing subsequence of different types of random sequences.&lt;br&gt; This very simple model is connected to statistical physics via its&lt;br&gt; relation to the Kardar-Parisi-Zhang universality class, which describes&lt;br&gt; the fluctuations of the surface of many growth processes.&lt;br&gt; For a case with known asymptotic distribution of the length we can show&lt;br&gt; a convergence of our measured distributions to the asymptotic form for&lt;br&gt; very large parts of the distribution. For another case we can confirm&lt;br&gt; a proposed scaling law also in the far tails of the distribution.&lt;/p&gt; &lt;p&gt;The fourth project of this thesis takes a look at the robustness of&lt;br&gt; networks. Since all systems of interacting objects, be it social&lt;br&gt; networks, energy grids or theoretical models on grids or more&lt;br&gt; complicated topologies, can be modeled with networks, it is of&lt;br&gt; fundamental interest how robust these systems are to failures of single&lt;br&gt; objects. Therefore we looked at a rather simple property of networks,&lt;br&gt; the size of the largest biconnected component. The biconnected component&lt;br&gt; is invulnerable to failures of one single object, such that a large&lt;br&gt; biconnected component is an indication for a robust network. We studied&lt;br&gt; the distribution of its size for two otherwise very well studied&lt;br&gt; network models.&lt;/p&gt;</dct:description>
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