Random Walks in Noninteger Dimension

Document Type

Article

Publication Date

1-1-1994

Department

Physics and Astronomy

School

Mathematics and Natural Sciences

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 D‐dimensional 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 number‐theoretic interpretation.

Publication Title

Journal of Mathematical Physics

Volume

35

Issue

1

First Page

368

Last Page

388

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