We show that one-dimensional random walks in a quasi-periodic environment with Liouville frequency generically have an erratic statistical behavior. In the recurrent case we show that neither quenched nor annealed limit theorems hold and both drift and variance exhibit wild oscillations, being logarithmic at some times and almost linear at other times. In the transient case we show that the annealed Central Limit Theorem fails generically. These results are in stark contrast with the Diophantine case where the Central Limit Theorem with linear drift and variance was established by Sinai.
Maria Saprykina was supported in part by the Swedish Research Council, VR 2015-04012. Dmitry Dolgopyat was supported in part by the NSF. Bassam Fayad was supported in part by ANR-15-CE40-0001 grant and Knut and Alice Wallenberg foundation, grant KAW 2016.0403.
We are grateful for two anonymous referees who made numerous comments and suggestions that helped us make substantial revisions to the first version of this paper. Bassam Fayad thanks the KTH for excellent working conditions during his visit.
"Erratic behavior for 1-dimensional random walks in a Liouville quasi-periodic environment." Electron. J. Probab. 26 1 - 36, 2021. https://doi.org/10.1214/21-EJP622