Study of the shape of random walks. II. Inertia moment ratios and the two-dimensional asphericity

Abstract

A study of some features of the quantities used to describe the shape of random walks, focusing on the probability distribution of the asphericity of discrete random walks in two dimensions, is presented. A connection is established between the asphericity distribution and the probability distribution corresponding to the ratio of the principal inertia moments. The probability distributions of arbitrary inertia moment ratios are analysed for varying spatial dimension, and an analytic expression for them is found which presents excellent agreement with Monte Carlo data in all the cases considered. This function is then used to obtain the corresponding two-dimensional asphericity distribution, which also proved to be a very good approximation to the true distribution obtained from an independent Monte Carlo simulation.