Abstract
We continue our investigation of a model of random walks on lattices with two kinds of points, “black” and “white.” The colors of the points are stochastic variables with a translation-invariant, but otherwise arbitrary, joint probability distribution. The steps of the random walk are independent of the colors. We are interested in the stochastic properties of the sequence of consecutive colors encountered by the walker. In this paper we first summarize and extend our earlier general results. Then, under the restriction that the random walk be symmetric, we derive a set of rigorous inequalities for the average length of the subwalk from the starting point to a first black point and of the subwalks between black points visited in succession. A remarkable difference in behavior is found between subwalks following an odd-numbered and subwalks following an evennumbered visit to a black point. The results can be applied to a trapping problem by identifying the black points with imperfect traps.
Similar content being viewed by others
References
W. Th. F. den Hollander and P. W. Kasteleyn,Physica 117A:179 (1983).
W. Feller,An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. (Wiley, New York, 1968) and Vol. 2, 2nd ed. (Wiley, New York, 1971).
F. Spitzer,Principles of Random Walk, 2nd ed. (Springer, New York, 1976).
E. Seneta,Non-negative Matrices and Markov Chains, 2nd ed. (Springer, New York, 1980).
D. R. Cox and H. D. Miller,The Theory of Stochastic Processes (Methuen, London, 1965).
M. Marcus and H. Minc,A Survey of Matrix Theory and Matrix Inequalities (Allyn and Bacon, Boston, 1964).
E. F. Beckenbach and R. Bellman,Inequalities (Springer, Berlin, 1961).
E. W. Montroll,Proc. Symp. Appl. Math. 16:193 (1964).
M. Parodi,La Localisation des Valeurs Caractéristiques des Matrices et ses Applications (Gauthiers-Villars, Paris, 1959).
H. R. Pitt,Proc. Cambridge Philos. Soc. 38:325 (1942).
X. X. Nguyen,Z. Wahrscheinlichkeitstheorie verw. Gebiete 48:159 (1979).
M. A. Akcoglu and U. Krengel,J. Reine Angew. Math. 323:53 (1981).
C. J. Preston,Gibbs States on Countable Sets, Cambridge Tracts in Mathematics 68 (Cambridge University Press, Cambridge, 1974).
P. W. Kasteleyn and W. Th. F. den Hollander,J. Stat. Phys. 30:363 (1983).
P. R. Halmos,Measure Theory (Springer, New York, 1974).
G. H. Weiss and R. J. Rubin,Adv. Chem. Phys. 52:363 (1982).
Proceedings of a Symposium on Random Walks, U.S. National Bureau of Standards, Gaithersburg, Maryland, 1982, inJ. Stat. Phys.,30:(2) (1983).
W. Th. F. den Hollander and P. W. Kasteleyn,Physica 112A:523 (1982).
E. W. Montroll,J. Phys. Soc. Japan Suppl. 26:6 (1969).
W. Th. F. den Hollander,J. Stat. Phys. 37:331 (1984).
H. B. Rosenstock,J. Math. Phys. 21:1643 (1980).
L. Breiman,Probability (Addison-Wesley, Reading, Massachusetts, 1968).
H. C. P. Berbee, Random walks with stationary increments and renewal theory (thesis),Mathematical Centre Tracts 112 (Amsterdam, 1979).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
den Hollander, W.T.F., Kasteleyn, P.W. Random walks on lattices with points of two colors. II. Some rigorous inequalities for symmetric random walks. J Stat Phys 39, 15–52 (1985). https://doi.org/10.1007/BF01007973
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01007973