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

In the thesis at hand Monte Carlo methods originating from statistical
physics are applied to study various problems in far more detail than
before. While all those problems have in common that they were up to
now mainly studied in regards to the mean values of some observable, in
this thesis the full distribution including very rare events with
probabilities in the order of 10^-100 and smaller are obtained and
discussed.

The first and largest project of this thesis is about the distribution
of the volume and surface of the convex hulls around the traces of
random walks. The first part of this project looks at the hulls of
standard random walks. For this rather simple model much progress was
made in the last decades and it is the only problem of this thesis for
which prior numerical results of the whole distribution exists in the special
case of two dimensions. Therefore this thesis focuses on a generalization
to higher dimensions. The second part of this project scrutinizes more
complicated types of random walks which interact with their past
trajectory. This interaction makes these random walks suitable as models
for, e.g., polymers. The same interaction also leads to an increased
difficulty in obtaining results analytically, such that the numerical
examination of the whole distribution seems worthwhile.

The second project examines the distribution of the ground-state energy
of a generalized random-energy model, a toy model from statistical
physics with applications to phase transitions and spin glasses.
There we find a universal asymptotic form for the distribution of the
ground-state energies in the limit of large systems, only dependent via
two parameters on the behavior of the underlying distribution of
the single energy levels in the system.

The third project scrutinizes the distribution of the length of the
longest increasing subsequence of different types of random sequences.
This very simple model is connected to statistical physics via its
relation to the Kardar-Parisi-Zhang universality class, which describes
the fluctuations of the surface of many growth processes.
For a case with known asymptotic distribution of the length we can show
a convergence of our measured distributions to the asymptotic form for
very large parts of the distribution. For another case we can confirm
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
networks. Since all systems of interacting objects, be it social
networks, energy grids or theoretical models on grids or more
complicated topologies, can be modeled with networks, it is of
fundamental interest how robust these systems are to failures of single
objects. Therefore we looked at a rather simple property of networks,
the size of the largest biconnected component. The biconnected component
is invulnerable to failures of one single object, such that a large
biconnected component is an indication for a robust network. We studied
the distribution of its size for two otherwise very well studied
network models.