Thesis Open Access

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

Schawe, Hendrik


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    <subfield code="a">Hartmann, Alexander K.</subfield>
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    <subfield code="u">Universität zu Köln</subfield>
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    <subfield code="a">Krug, Joachim</subfield>
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    <subfield code="a">&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;</subfield>
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    <subfield code="a">10.1103/PhysRevE.96.062101</subfield>
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