Random Walks in Noninteger Dimension

Details Random Walks in Noninteger Dimension Random Walks in Noninteger Dimension

1993-11-10

arxiv.org/abs/hep-lat/9311011v1

J.Math.Phys. 35 (1994) 368-388

Physics

High Energy Physics

High Energy Physics - Lattice

25 pages, 5 figures included, 2 figures on request, plain TEX

Scientific paper

10.1063/1.530778

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 we propose a random walk 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. We give the exact result 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.

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