Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T21:27:13.562Z Has data issue: false hasContentIssue false
  • English
  • Français

Return probabilities for certain three-dimensional random walks

Published online by Cambridge University Press:  14 July 2016

D. J. Daley
D. J. Daley*
Affiliation:
The Australian National University, Canberra
*
∗Postal address: Statistics Department (IAS), The Australian National University, P.O. Box 4, Canberra A.C.T. 2600, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

For some three-dimensional random walks on the cubic lattice, the probability of the walk returning to its starting point is given numerically.

Keywords

RANDOM WALKRETURN PROBABILITYCOMPLETE ELLIPTIC INTEGRAL
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 1 , March 1979 , pp. 45 - 53
Copyright
Copyright © Applied Probability Trust 1979 

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

Work done while the author was Visiting Professor, Statistics Department, University of Wyoming.

References

References

Byrd, P. F. and Friedman, M. D. (1971) Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn, revised. Springer–Verlag, New York.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Lehman, R. S. (1951) A problem on random walk. Proc. 2nd Berkeley Symp. Math. Statist. Prob., 263–268.Google Scholar
Montroll, E. W. (1956a) Random walks in multi-dimensional spaces, especially on periodic lattices. SIAM J. 4, 241–260.Google Scholar
Montroll, E. W. (1956b) Theory of the vibration of simple cubic lattices with nearest neighbor interactions. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 3, 209–246.Google Scholar
Montroll, E. W. (1964) Random walks on lattices. Proc. Symp. Appl. Math. 16, 193–220.CrossRefGoogle Scholar
Moran, P. A. P. (1968) Introduction to Probability Theory. Clarendon Press, Oxford.Google Scholar
Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, Princeton, N.J.CrossRefGoogle Scholar
Watson, G. N. (1939) Three triple integrals. Quart. J. Maths (Oxford) 10, 266–276.Google Scholar

Additional References

Barber, M. N. and Ninham, B. W. (1970) Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York.Google Scholar
Carazza, B. (1977) The history of the random-walk problem: considerations on the interdisciplinarity in modern physics. Riv. Nuovo Cimento 7, 419–427.Google Scholar
Domb, C. and Fisher, M. E. (1958) On random walks with restricted reversals. Proc. Camb. Phil. Soc. 54, 48–59.CrossRefGoogle Scholar
Glasser, M. L. and Wood, V. E. (1971) A closed form evaluation of the elliptic integral. Math. Comp. 25, 535–536.Google Scholar
Glasser, M. L. and Zucker, I. J. (1977) Extended Watson integrals for the cubic lattices. Proc. Natn. Acad. Sci. USA 74, 1800–1801.Google Scholar
Glasser, M. L. and Zucker, I. J. (1979) In Theoretical Chemistry: Advances and Perspectives, Vol. 5, ed. Eyring, H. and Henderson, D. Academic Press, New York.Google Scholar
Koiwa, M. (1977) Random walks on three-dimensional lattices: a matrix method for calculating the probability of eventual return. Phil. Mag. 36, 893–905.CrossRefGoogle Scholar
Novoseller, D. E. and Glasser, M. L. (1977) A combinatorial identity. J. Combinatorial Theory A 22, 372–377.Google Scholar