Random walks in noninteger dimension

Journal Article · Sat Jan 01 04:00:00 EST 1994 · Journal of Mathematical Physics (New York); (United States)

DOI:https://doi.org/10.1063/1.530778· OSTI ID:5289162

Bender, C M; Boettcher, S ^[1]; Mead, L R ^[2]

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1. Department of Physics, Washington University, St. Louis, Missouri 63130-4899 (United States)
2. Department of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, Mississippi 39406-5046 (United States)

One can define a random walk on a hypercubic lattice in a space of integer dimension [ital 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 [ital D]. However, these formulas are unacceptable as probabilities when continued to noninteger [ital 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 [ital D]. This [ital 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 [ital D] in terms of the Riemann zeta function. This result has a number-theoretic interpretation.

• OSTI ID: 5289162
• Journal Information: Journal of Mathematical Physics (New York); (United States), Journal Name: Journal of Mathematical Physics (New York); (United States) Vol. 35:1; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

661300* -- Other Aspects of Physical Science-- (1992-)
662110 -- General Theory of Particles & Fields-- Theory of Fields & Strings-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
CRYSTAL LATTICES
CRYSTAL STRUCTURE
CUBIC LATTICES
FIELD THEORIES
FUNCTIONS
GREEN FUNCTION
LATTICE FIELD THEORY
MANY-DIMENSIONAL CALCULATIONS
PERTURBATION THEORY
PROBABILITY
QUANTUM FIELD THEORY
SPHERES
STOCHASTIC PROCESSES