Abstract
We give an elementary proof that the second coordinate (the scenery process) of theT, T −1-process associated to any mean zero i.i.d. random walk onZ d is not a finitary factor of an i.i.d. process. In particular, this yields an elementary proof that the basicT, T −1-process is not finitarily isomorphic to a Bernoulli shift (the stronger fact that it is not Bernoulli was proved by Kalikow). This also provides (using past work of den Hollander and the author) an elementary example, namely theT, T −1-process in 5 dimensions, of a process which is weak Bernoulli but not a finitary factor of an i.i.d. process. An example of such a process was given earlier by del Junco and Rahe. The above holds true for arbitrary stationary recurrent random walks as well. On the other hand, if the random walk is Bernoulli and transient, theT, T −1-process associated to it is also Bernoulli. Finally, we show that finitary factors of i.i.d. processes with finite expected coding volume satisfy certain notions of weak Bernoulli in higher dimensions which have been previously introduced and studied in the literature. In particular, this yields (using past work of van den Berg and the author) the fact that the Ising model is weak Bernoulli throughout the subcritical regime.
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References
H. C. P. Berbee,Random Walks with Stationary Increments and Renewal Theory, Ph.D. thesis, Amsterdam, 1979.
J. van den Berg and J. E. Steif,On the existence and non-existence of finitary codings for a class of random fields, The Annals of Probability27 (1999), 1501–1522.
R. M. Burton and J. E. Steif,Quite weak Bernoulli with exponential rate and percolation for random fields, Stochastic Processes and their Applications58 (1995), 35–55.
R. M. Burton and J. E. Steif,Coupling surfaces and weak Bernoulli in one and higher dimensions, Advances in Mathematics132 (1997), 1–23.
F. M. Dekking,On transience and recurrence of generalized random walks, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete61 (1982), 459–465.
M. D. Donsker and S. R. S. Varadhan,On the number of distinct sites visited by a random walk, Communications on Pure and Applied Mathematics32 (1979), 721–747.
R. Durrett,Probability: Theory and Examples (2nd edition), Wadsworth Publ. Co., Duxbury Press, Belmont (CA), 1996.
O. Häggström,Coloring percolation clusters at random, submitted.
Y. Hamana and H. Kesten,A large deviation result for the range of random walk and for the Wiener sausage, Probability Theory and Related Fields, to appear.
C. Hoffman,The scenery of the T, T −1 is not Bernoulli, submitted.
F. den Hollander and J. E. Steif,Mixing properties of the generalized T, T −1-process, Journal d’Analyse Mathématique72 (1997), 165–202.
A. del Junco and M. Rahe,Finitary codings and weak Bernoulli partitions, Proceedings of the American Mathematical Society75 (1979), 259–264.
S. Kalikow,T, T −1 transformation is not loosely Bernoulli, Annals of Mathematics115 (1982), 393–409.
K. Marton and P. Shields,The positive-divergence and blowing-up properties, Israel Journal of Mathematics86 (1994), 331–348.
D. S. Ornstein,Ergodic Theory, Randomness and Dynamical Systems, Yale University Press, New Haven, 1974.
M. Smorodinsky and J-P. Thouvenot,Bernoulli factors that span a transformation, Israel Journal of Mathematics32 (1979), 39–43.
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Steif, J.E. TheT, T −1-process, finitary codings and weak Bernoulli. Isr. J. Math. 125, 29–43 (2001). https://doi.org/10.1007/BF02773372
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DOI: https://doi.org/10.1007/BF02773372