Distributions of convex hulls of random walks

随机游动是物理、生物和社会过程的普遍模型。 在许多情况下，由一个或多个步行者覆盖的区域是特别感兴趣的，例如，when describing描述home ranges范围of anmials动物. 使用最小凸多边形，称为凸壳，与动物的踪迹接壤，是描述家域的一种简单而有效的方法，可用于任何类型的动物。（随机行走）数据。拟议的项目涉及数值研究性质的凸壳的不同类型的随机行走模型，从简单的在不同的维度，通过随机行走的集合到描述相互作用的动物在其栖息地中的运动的行走。典型的属性，如凸面外壳的周长和面积的平均值，很容易在数值上获得，在少数情况下，它们也可以从分析研究中获得。然而，一个随机过程的精确描述涉及到全分布的知识，这是迄今为止我们考虑研究的任何模型都没有得到的。 因此，为了在大范围的支持下对这些分布进行彻底的分析，我们必须能够对非常小的概率进行采样。 因此，本项目的目标是应用复杂的大偏差算法来研究不同随机游走类型的凸包的分布，直到非常小的概率，如10^-300。这些分布可以通过数据拟合与标准分布（如极值分布）进行比较。更好的是，考虑到不同的系统尺寸，有限尺寸修正项可以被系统地表征和分析。

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.