The shapes of open and closed random walks: a 1/d expansion

Abstract

A new technique for calculating the shapes of random walks is presented. The method is used to derive an exact analytical expression for the asphericity of an unrestricted closed or ring walk embedded in d spatial dimensions. A graphical procedure is developed to systematise a 1/d series expansion for the individual principal radii of gyration and their respective probability distribution functions P(R²_i)(12) for both open and closed walks and selected terms in the 1/d expansion are summed to all orders in 1/d in the determination of P(R²_i). This leads to an explicit analytical form for P(R²_i) for open walks. The distribution of the largest eigenvalue is compared with a distribution obtained from numerical simulations of walks in three dimensions. The agreement between the two is extremely good. Other predictions for various parameters that characterise the average shape of open and closed walks in three dimensions are also found to agree remarkably well with the results of simulations, the error being of the order of 5%.