Shapes and sizes of arbitrary random walks at O(1/d3) II. Asphericity and prolateness parameters

For arbitrary random walks in any d-dimensional space, expansions in powers of 1/d of asphericity and prolateness parameters and moments of the inverse size ratio have been developed, which, at O(1/d3), yield very good approximations to exact values of the parameters for chains, rings, dumbbells and 3- and 5-arm regular stars. The 1/d-expansions have also been used to obtain an estimate of these shape asymmetry parameters for 3D Edwards chains, rings, dumbbells and 3-arm stars and to give a mathematical proof that infinitely large random nets such as Bethe lattice or starburst and Mckayʹs net exhibit spherical symmetry. For arbitrary random walks at d = ∞, it is proved that these parameters coincide with their corresponding factors, while for an end-looped self-avoiding walk, it is found that its shape asymmetry is even larger than that of an open SAW. An 1/f-expansion of the parameters for f-arm regular stars has also been obtained, and a comparison of the dimensionality dependence of the parameters with that of the corresponding factors has been made for the four types of random walks.