Distributions of convex hulls of random walks

• Applicant: Professor Dr. Alexander Hartmann
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2014 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 257398711

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.

• DFG Programme: Research Grants
• International Connection: France
• Participating Person: Professor Dr. Satya Majumdar