SPHERICALLY SYMMETRICAL RANDOM-WALKS IN NONINTEGER DIMENSION

A previous article proposed a new kind of random walk on a spherically symmetric lattice in arbitrary noninteger dimension D. Such a lattice avoids the problems associated with a hypercubic lattice in noninteger dimension. This article examines the nature of spherically symmetric random walks in detail. A large-time asymptotic analysis of these random walks is performed and the results are used to determine the Hausdorff dimension of the process. Exact results are Obtained in terms of Hurwitz functions (incomplete zeta functions) for the probability of a walker going from one region of the spherical lattice to another. Finally, it is shown that the probability that the paths of K independent random walkers will intersect vanishes in the continuum limit if D > 2K/(K-1).