Abstract
Some fifteen years ago, Shuler formulated three conjectures relating to the large-time asymptotic properties of a nearest-neighbor random walk on ℤ2 that is allowed to make horizontal steps everywhere but vertical steps only on a random fraction of the columns. We give a proof of his conjectures for the situation where the column distribution is stationary and satisfies a certain mixing codition. We also prove a strong form of scaling to anisotropic Brownian motion as well as a local limit theorem. The main ingredient of the proofs is a large-deviation estimate for the number of visits to a random set made by a simple random walk on ℤ. We briefly discuss extensions to higher dimension and to other types of random walk.
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References
K. E. Shulet,Physica 95A:12–34 (1979).
E. W. Montroll and G. H. Weiss,J. Math. Phys. 6:176–181 (1965).
A. de Masi, P. A. Ferrari, S. Goldstein, and D. W. Wick,J. Stat. Phys. 55: 787–855 (1989).
H. Kesten,Ann. Inst. H. Poincaré 22:425–487 (1986).
H. Silver, K. E. Shuler, and K. Lindenberg, inStatistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum Press, New York, 1977), pp. 463–505.
V. Seshadri, K. Lindenberg, and K. E. Shuler,J. Stat. Phys. 21:517–548 (1979).
M. Westcott,J. Stat. Phys. 27:75–82 (1982).
C. C. Heyde,J. Stat. Phys. 27:721–730 (1982).
C. C. Heyde, M. Westcott, and E. R. Williams,J. Stat. Phys. 28:375–380 (1982).
P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).
J. B. T. M. Roerdink and K. E. Shulet,J. Stat. Phys. 40:205–240 (1985).
J. B. T. M. Roerdink and K. E. Shuler,J. Stat. Phys. 41:581–606 (1985).
J. B. T. M. Roerdink,Physica 132A:253–268 (1985).
D. Ruelle,Thermodynamic Formalism (Addison-Wesley, Reading, Massachusetts, 1978).
M. D. Donsker and S. R. S. Varadhan,Commun. Pure Appl. Math. 28:721–747 (1975).
H. Kesten and F. Spitzer,Z. Wahrsch. Verw. Gebiete 50:5–25 (1979).
S. Kakutani, inProceedings 2nd Berkeley Symposium on mathematics, Statistics, and Probability (University of California Press, Berkeley, 1951), pp. 247–261.
J. Komlós, P. Major, and G. Tusnády,Z. Wahrsch. Verw. Gebiete 32:111–131 (1975);34:33–58 (1976).
L. Breiman,Probability (Addison-Wesley, Reading, Massachusetts, 1968).
M. Reed and B. Simon,Methods of Modern Mathematical Physics, I: Functional Analysis (Academic Press, New York, 1972).
J. Neveu,Ann. Inst. Fourier (Grenoble)22:85–130 (1972).
N. H. Bingham, C. M. Goldie, and J. L. Teugels,Regular Variation (Cambridge University Press, Cambridge, 1987).
D. V. Widder,The Laplace Transform (Princeton University Press, Princeton, New Jersey, 1941).
F. Spitzer,Principles of Random Walk, 2nd ed. (Springer, New York, 1976).
F. den Hollander, J. Naudts, and F. Redig, preprint UIA Antwerpen (1993), to appear inJ. Stat. Phys.
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Dedicated to Prof. K. E. Shuler on the occasion of his 70th birthday, celebrated at a Symposium in his honor on July 13, 1992, at the University of California at San Diego, La Jolla, California.
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den Hollander, F. On three conjectures by K. E. Shuler. J Stat Phys 75, 891–918 (1994). https://doi.org/10.1007/BF02186749
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DOI: https://doi.org/10.1007/BF02186749