Distributions of convex hulls of random walks

随机行走是物理、生物和社会过程中普遍存在的模型。在许多情况下，一个或多个步行者所覆盖的区域是特别重要的，例如，当描述动物的家园范围时。用最小凸多边形(称为凸壳)来描述动物的轨迹是一种简单而通用的方法，可以用来描述任何类型的(随机游动)数据。拟议的项目涉及数值研究不同类型随机行走模型的凸壳的性质，从不同维度的简单模型，到描述动物在其栖息地中相互作用的行走的集合。典型的性质，如凸壳的周长和面积的平均值，在数值上很容易获得，在极少数情况下，它们也可以从分析研究中获得。然而，对随机过程的全面描述涉及到完全分布的知识，对于我们考虑在这里研究的任何模型，到目前为止还没有得到这一知识。因此，要在很大的支持范围内对这些分布进行全面分析，我们必须能够抽样非常小的概率。因此，这个项目的目的是应用复杂的大偏差算法来研究不同随机游动类型的凸壳的分布，小到10^-300的概率。这些分布可以通过与标准分布(如极值分布)的数据拟合来比较。更好的是，考虑到不同的系统大小，可以系统地刻画和分析有限大小的修正项。

Random walks are ubiquitous models for physical, biological and socialprocesses. In many circumstances the area covered by one or multiplewalkers is of particular interest, e.g., when describing home rangesof anmials. The usage of minimum convex polygons, called convexhulls, bordering the trace of an animal is a simple yet versatileway to describe the home range and can be used for any type of(random-walk) data.The proposed project involves numerically studying properties of theconvex hulls of different types of random walk models, from simpleones in different dimensions, over ensembles of random walks to walksdescribing the movement of interacting animals in their habitats.Typical properties like the average of perimeter and area of theconvex hulls are numerically easily to obtain and in few cases theyare also available from analytical studies. Nevertheless, acomprehensive description of a random process involves the knowledgeof the full distribution, which has not been obtained so far for anyof the models we consider to investigate here. Therefore, for athorough analysis of these distributions over a large range of thesupport, we must be able to sample very small probabilities. Thus, itis the aim of this project to apply sophisticated large-deviationsalgorithms to study the distributions of convex hulls of differentrandom-walk types down to very small probabilities like 10^-300.These distributions can be compared via data fitting with standarddistributions, like extreme-value distributions. Even better,considering different system sizes, finite-size correction terms canbe characterized and analyzed systematically.