RANDOM-WALKS IN NONINTEGER DIMENSION

One can define a random walk on a hypercubic lattice in a space of int eger dimension D. For such a process formulas can be derived that expr ess the probability of certain events, such as the chance of returning to the origin after a given number of time steps. These formulas are physically meaningful for integer values of D. However, these formulas are unacceptable as probabilities when continued to noninteger D beca use they give values that can be greater than 1 or less than 0. In thi s paper a different kind of random walk is proposed which gives accept able probabilities for all real values of D. This D-dimensional random walk is defined on a rotationally symmetric geometry consisting of co ncentric spheres. The exact result is given for the probability of ret urning to the origin for all values of D in terms of the Riemann zeta function. This result has a number-theoretic interpretation.