Random walks in noninteger dimension
Abstract
One can define a random walk on a hypercubic lattice in a space of integer dimension D. For such a process formulas can be derived that express 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 because they give values that can be greater than 1 or less than 0. In this paper a different kind of random walk is proposed which gives acceptable probabilities for all real values of D. This Ddimensional random walk is defined on a rotationally symmetric geometry consisting of concentric spheres. The exact result is given for the probability of returning to the origin for all values of D in terms of the Riemann zeta function. This result has a numbertheoretic interpretation.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 January 1994
 DOI:
 10.1063/1.530778
 arXiv:
 arXiv:heplat/9311011
 Bibcode:
 1994JMP....35..368B
 Keywords:

 High Energy Physics  Lattice
 EPrint:
 25 pages, 5 figures included, 2 figures on request, plain TEX