SHAPES AND SIZES OF ARBITRARY RANDOM-WALKS AT O(1 D(3)) .1. 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 s(2 ) divided by [s(2)] of open random walks, has been developed, which, a t O(1/d(3)), yields a very good approximation to the exact value for c hains (d greater than or equal to 2) and rings (d greater than or equa l to 1), and for the first time, gives an estimate of the most probabl e size ratio for end-looped random walks. Asymptotic distribution func tions for large and small size ratio have also been investigated analy tically for open and closed random walks with explicit results given u p to the fourth order for any values of d. For random walks at d = inf inity, it has been proved that the most probable size coincides with m ean size and the alpha th shape factor is inversely proportional to th e alpha th eigenvalue of the architecture matrix for the walks.