Erratic behavior for 1-dimensional random walks in a Liouville quasi-periodic environment

2019/01/30 by Dolgopyat, Dmitry, Bassam Fayad, Fayad, Bassam +3
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #37E45 #60G50 #Diffusion and Search Dynamics #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.1901.10709

openalex publication_date 2019/01/30 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28

Abstract

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.

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