Asymmetric random walk on a random Thue-Morse lattice

Abstract

We study the behavior of an asymmetric random walk in a one-dimensional environment whose nonuniformity is in between that of quasi-periodic and random. We construct the environment from arithmetic subsequences of the Thue-Morse sequence. The construction induces in a natural way a measure μ on the space of environments which is invariant and ergodic with respect to translations but is not mixing and has zero entropy. The behavior of the random walk is rather similar to that found by Sinai for the Bernoulli case, when μ is a product measure for which the entropy has its maximum value; i.e. the particle motion is subdiffusive, the displacement growing in time as $\text{(}\text{log}\text{)t)}{}^{\mathrm{<mtext>1</mtext><mtext>\beta </mtext>}}\text{, β =}\text{log}\text{3/}\text{log}\text{4}$. The nature of the dramatic Sinai-Golosov “localization” is however quite different, exhibiting an interesting fractal structure whose nature depends upon the time scale of observation.