Large Deviations of Convex Hulls of Random Walks and Other Stochastic Models
Creators: name: Schawe, Hendrik
Publisher: name: Zenodo
Year: 2019

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