Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T08:16:33.741Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Berger, M. A. (1987) The random walk winding number problem: convergence to a diffusion process with excluded area. J. Phys. A: Math. Gen. 20, 5949–5960.Google Scholar
Berger, M. A. (1988) The development of structure in coronal magnetic fields. In Activity in Cool Star Envelopes, ed. Havnes, O., Reidel, Dordrecht.Google Scholar
Chandrasekhar, S. (1943) Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89. Reprinted in Wax (1954).Google Scholar
Gold, T. (1964) In The Physics of Solar Flares, (ed. Hess, W.), NASA Publ. SP-50.Google Scholar
Levy, P. (1940) Le mouvement Brownién plan. Amer. J. Math. 62, 487–550.Google Scholar
Levy, P. (1965) Processus Stochastique et Mouvement Brownién. Gauthier-Villars, Paris. See Chapter VII.Google Scholar
Lyons, T. J. and Mckean, H. P. (1984) Winding of plane Brownian motion. Adv. Math. 51, 212–225.Google Scholar
Parker, E. N. (1972) Topological dissipation and the small-scale fields in turbulent gases. Astrophys. J. 174, 499–510.Google Scholar
Parker, E. N. (1979) Cosmical Magnetic Fields, their Origin and their Activity. Clarendon Press, Oxford.Google Scholar
Parker, E. N. (1983) Magnetic neutral sheets in evolving fields. II Formation of the solar corona. Astrophys. J. 264, 642–647.Google Scholar
Pearson, K. (1905a) The problem of random walk. Nature 72, 294.CrossRefGoogle Scholar
Pearson, K. (1905b) The problem of random walk. Nature 72, 342.CrossRefGoogle Scholar
Pearson, K. (1906) Mathematical Contributions to the Theory of Evolution–XV. A Mathematical Theory of Random Migration. Draper&s Company Research Memoirs, Biometric Series. (Memoir written with the assistance of John Blakeman.) Cambridge University Press, London.Google Scholar
Pitman, J. W. and Yor, M. (1984) The asymptotic joint distribution of planar Brownian motion. Bull. Amer. Math. Soc. 10, 109–111.CrossRefGoogle Scholar
Rayleigh, Lord (Strutt, J. W.) (1919) On the problem of random vibrations, and of random flights in one, two or three dimensions. Phil. Mag. (Ser. 6) 37, 321–347. See also Collected Papers 6, 604-626. (Dover, New York, 1964).Google Scholar
Roberts, P. H. and Ursell, H. D. (1960) Random walk on a sphere and on a Riemannian manifold. Phil. Trans. R. Soc. London A252, 317–356.Google Scholar
Roberts, P. H. and Winch, D. E. (1984) On random rotations. Adv. Appl. Prob. 16, 638–655.Google Scholar
Spitzer, F. (1958) Some theorems concerning 2-dimensional Brownian motion. Amer. Math. Soc. Trans. 87, 187–197.Google Scholar
Watson, G. N. (1944) A Treatise on the Theory of Bessel Functions. Cambridge University Press, London.Google Scholar
Wax, N., (Ed) (1954) Selected Papers on Noise and Stochastic Processes. Dover, New York.Google Scholar