Hostname: page-component-7bb8b95d7b-dtkg6 Total loading time: 0 Render date: 2024-09-17T23:27:05.922Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattice-valued random walks with Markov chain dependent steps

Published online by Cambridge University Press:  24 October 2008

D. J. Daley
D. J. Daley
Affiliation:
Statistics Department (IAS), The Australian National University
Get access Rights & Permissions [Opens in a new window]

Abstract

The probability of ever returning to the origin and the mean square displacement after n steps are studied for some lattice-valued random walks, whose successive steps constitute a Markov chain on a finite state space with transition probabilities of a simple kind, and such that the returns to the origin form a regenerative phenomenon. The case of walks on a diamond lattice with no immediate reversals is included: this example is relevant as a polymer chain building model. The numerical evaluation of the return probabilities of some three-dimensional walks is discussed and examples given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 1 , July 1979 , pp. 115 - 126
Copyright
Copyright © Cambridge Philosophical Society 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.)

References

REFERENCES

(1)Abramowitz, M. and Stegun, I. A. (eds.). Handbook of mathematical functions (National Bureau of Standards, Dover Reprint, 1964).Google Scholar
(2)Barber, M. N. and Ninham, B. W.Random and restricted walks: theory and applications (New York and London, Gordon and Breach, 1970).Google Scholar
(3)Daley, D. J. Return probabilities for certain three-dimensional random walks. J. Appl. Probability 16 (1979) (to appear).CrossRefGoogle Scholar
(4)Domb, C. and Fisher, M. E. On random walks with restricted reversals. Proc. Cambridge Philos. Soc. 54 (1958), 48–58.CrossRefGoogle Scholar
(5)Feller, W.An introduction to probability theory and its applications, vol. 2 (New York, Wiley, 1966).Google Scholar
(6)Flory, P. J.Statistical mechanics of chain molecules (New York, Wiley-Interscience, 1969).CrossRefGoogle Scholar
(7)Gillis, J. Correlated random walk. Proc. Cambridge Philos. Soc. 51 (1955), 639–651.CrossRefGoogle Scholar
(8)Glasser, M. L. and Zucker, I. J. Extended Watson integrals for the cubic lattices. Proc. Nat. Acad. Sci. USA 74 (1977), 1800–1801.CrossRefGoogle Scholar
(9)Gordon, M., Torkington, J. A. and Ross-Murphy, S. B.The graph-like state of matter. 9. Statistical thermodynamics of dilute polymer solutions. Macromolecules 10 (1977), 1090–1100.CrossRefGoogle Scholar
(10)Jain, N. C. and Pruitt, W. E.The range of transient random walk. J. Analyse Math. 24 (1971), 369–393.CrossRefGoogle Scholar
(11)Koiwa, M.Random walks on three-dimensional lattices: a matrix method for calculating the probability of eventual return. Philos. Mag. 36 (1977), 893–905.CrossRefGoogle Scholar
(12)Lehman, R. S.A problem on random walk. Proc. Second Berkeley Symp. Math. Stat. Probab. (1951), 263–268.Google Scholar
(13)Montroll, E. W.Theory of the vibration of simple cubic lattices with nearest neighbour interactions. Proc. Third Berkeley Symp. Math. Stat. Probab. (1956), Vol. III, 209–246.Google Scholar
(14)Montroll, E. W. and Weiss, G. H.Random walks on lattices. II. J. Mathematical Phys. 6 (1965), 167–181.CrossRefGoogle Scholar
(15)Spitzer, F.Principles of random walk (Princeton, Van Nostrand, 1964).CrossRefGoogle Scholar
(16)Watson, G. N., Three triple integrals. Quart. J. Math. (Oxford) 10 (1939), 266–276.CrossRefGoogle Scholar