Sinai’s random walk in random environment shows interesting patterns on the exponential time scale. We characterize the patterns that appear on infinitely many time scales after appropriate rescaling (a functional law of iterated logarithm). The curious rate function captures the difference between one-sided and two-sided behavior.
"Patterns in Sinai’s walk." Ann. Probab. 41 (3B) 1900 - 1937, May 2013. https://doi.org/10.1214/11-AOP724