Shapes and sizes of arbitrary random walks at O(1/d3) I. Most probable size ratio

For arbitrary random walks in any d-dimensional space, a 1/d expansion of the most probable size ratio, i.e., squared radius of gyration s2 divided by s2 of open random walks, has been developed, which, at O(1/d3), yields a very good approximation to the exact value for chains (d 2) and rings (d 1), and for the first time, gives an estimate of the most probable size ratio for end-looped random walks. Asymptotic distribution functions for large and small size ratio have also been investigated analytically for open and closed random walks with explicit results given up to the fourth order for any values of d. For random walks at d = ∞, it has been proved that the most probable size coincides with mean size and the αth shape factor is inversely proportional to the αth eigenvalue of the architecture matrix for the walks.