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 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.
- Publication:
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Journal of Mathematical Physics
- Pub Date:
- January 1994
- DOI:
- 10.1063/1.530778
- arXiv:
- arXiv:hep-lat/9311011
- Bibcode:
- 1994JMP....35..368B
- Keywords:
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- High Energy Physics - Lattice
- E-Print:
- 25 pages, 5 figures included, 2 figures on request, plain TEX