Spherically symmetric random walks. I. Representation in terms of orthogonal polynomials

Journal Article · Mon Jul 01 04:00:00 EDT 1996 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

DOI:https://doi.org/10.1103/PhysRevE.54.100· OSTI ID:285909

Bender, C M ^[1]; Cooper, F ^[2]; Meisinger, P N ^[1]

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1. Department of Physics, Washington University, St. Louis, Missouri 63130 (United States)
2. Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)

It is shown that, in general, a connection exists between orthogonal polynomials and semibounded random walks. This connection allows one to view a random walk as taking place on the set of integers that index the orthogonal polynomials. An illustration is provided by the case of spherically symmetric random walks. The correspondence between orthogonal polynomials and random walks enables one to express random-walk probabilities as weighted inner products of the polynomials. This correspondence is exploited to construct and analyze spherically symmetric random walks in {ital D}-dimensional space, where {ital D} is {ital not} restricted to be an integer. Such random walks can be described in terms of Gegenbauer (ultraspherical) polynomials. For example, Legendre polynomials can be used to represent the special case of two-dimensional spherically symmetric random walks. The weighted inner-product representation is used to calculate exact closed-form spatial and temporal moments of the probability distribution associated with the random walk. The polynomial representation of spherically symmetric random walks is then used to calculate the two-point Green{close_quote}s function for a rotationally symmetric free scalar quantum field theory. {copyright} {ital 1996 The American Physical Society.}

• OSTI ID: 285909
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics Journal Issue: 1 Vol. 54; ISSN 1063-651X; ISSN PLEEE8
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
PARTICLE PROPERTIES
POLYNOMIALS
RANDOM WALK
SPHERICAL CONFIGURATION
SYMMETRY
TRANSPORT THEORY