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

## TL;DR

This paper demonstrates that one-dimensional random walks in Liouville quasi-periodic environments exhibit highly irregular statistical behaviors, with standard limit theorems failing and oscillating drift and variance, contrasting with Diophantine cases.

## Contribution

It reveals the generic erratic behavior of random walks in Liouville environments, showing failure of classical limit theorems and contrasting with known Diophantine case results.

## Key findings

- Quenched and annealed limit theorems do not hold in recurrent Liouville environments.
- Drift and variance exhibit wild oscillations, logarithmic at times and linear at others.
- Annealed CLT fails in the transient Liouville environment.

## 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.