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On the winding number problem with finite steps

Published online by Cambridge University Press:  01 July 2016

M. A. Berger  and
P. H. Roberts
M. A. Berger
Affiliation:
High Altitude Observatory, Boulder
P. H. Roberts*
Affiliation:
University of California, Los Angeles
*
∗∗ Postal address: Department of Mathematics, University of California, Los Angeles, California 90024, USA.
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Abstract

The winding number problem (Lévy (1940)) concerns the net angle through which the route of a random walk winds about the origin. We consider the problem of finding the winding number for a walk with finite step sizes; the eigenfunction method (Roberts and Ursell (1960)) is shown to be inapplicable because the probability distribution for a sequence of steps of different length depends on the order in which those steps are taken. In the diffusion limit, however, commutivity is restored. We derive the winding number distribution for a diffusion process, starting from a point displaced from the origin, and consider its asymptotic form. An important difference between the finite step and diffusion distributions is that the former possesses finite moments while the latter does not. We compute numerically the finite step distributions for 20000 particles undergoing N = 100000 steps, and compare the results with the diffusion distribution. Even for small winding numbers, perceptible differences between the two distributions appear even for N as large as 100000.

Keywords

RANDOM WALKWINDING NUMBERSDIFFUSION DISTRIBUTIONSOLAR MAGNETISM
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 261 - 274
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

∗

Present address: Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland.

The National Center for Atmospheric Research is supported by the National Science Foundation.

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