Thesis Open Access

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

Schawe, Hendrik

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{
"publisher": "Zenodo",
"DOI": "10.5281/zenodo.3377932",
"language": "eng",
"title": "Large Deviations of Convex Hulls of Random Walks and Other Stochastic Models",
"issued": {
"date-parts": [
[
2019,
3,
19
]
]
},
"abstract": "<p>In the thesis at hand Monte Carlo methods originating from statistical<br>\nphysics are applied to study various problems in far more detail than<br>\nbefore. While all those problems have in common that they were up to<br>\nnow mainly studied in regards to the mean values of some observable, in<br>\nthis thesis the full distribution including very rare events with<br>\nprobabilities in the order of 10<sup>-100</sup> and smaller are obtained and<br>\ndiscussed.</p>\n\n<p>The first and largest project of this thesis is about the distribution<br>\nof the volume and surface of the convex hulls around the traces of<br>\nrandom walks. The first part of this project looks at the hulls of<br>\nstandard random walks. For this rather simple model much progress was<br>\nmade in the last decades and it is the only problem of this thesis for<br>\nwhich prior numerical results of the whole distribution exists in the special<br>\ncase of two dimensions. Therefore this thesis focuses on a generalization<br>\nto higher dimensions. The second part of this project scrutinizes more<br>\ncomplicated types of random walks which interact with their past<br>\ntrajectory. This interaction makes these random walks suitable as models<br>\nfor, e.g., polymers. The same interaction also leads to an increased<br>\ndifficulty in obtaining results analytically, such that the numerical<br>\nexamination of the whole distribution seems worthwhile.</p>\n\n<p>The second project examines the distribution of the ground-state energy<br>\nof a generalized random-energy model, a toy model from statistical<br>\nphysics with applications to phase transitions and spin glasses.<br>\nThere we find a universal asymptotic form for the distribution of the<br>\nground-state energies in the limit of large systems, only dependent via<br>\ntwo parameters on the behavior of the underlying distribution of<br>\nthe single energy levels in the system.</p>\n\n<p>The third project scrutinizes the distribution of the length of the<br>\nlongest increasing subsequence of different types of random sequences.<br>\nThis very simple model is connected to statistical physics via its<br>\nrelation to the Kardar-Parisi-Zhang universality class, which describes<br>\nthe fluctuations of the surface of many growth processes.<br>\nFor a case with known asymptotic distribution of the length we can show<br>\na convergence of our measured distributions to the asymptotic form for<br>\nvery large parts of the distribution. For another case we can confirm<br>\na proposed scaling law also in the far tails of the distribution.</p>\n\n<p>The fourth project of this thesis takes a look at the robustness of<br>\nnetworks. Since all systems of interacting objects, be it social<br>\nnetworks, energy grids or theoretical models on grids or more<br>\ncomplicated topologies, can be modeled with networks, it is of<br>\nfundamental interest how robust these systems are to failures of single<br>\nobjects. Therefore we looked at a rather simple property of networks,<br>\nthe size of the largest biconnected component. The biconnected component<br>\nis invulnerable to failures of one single object, such that a large<br>\nbiconnected component is an indication for a robust network. We studied<br>\nthe distribution of its size for two otherwise very well studied<br>\nnetwork models.</p>",
"author": [
{
"family": "Schawe, Hendrik"
}
],
"type": "thesis",
"id": "3377932"
}
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