Patterns in Sinai's walk

TL;DR

This paper studies the complex patterns of Sinai's random walk in random environments, revealing their behavior across multiple time scales and introducing a rate function that distinguishes between different directional behaviors.

Contribution

It characterizes the patterns in Sinai's walk on infinitely many time scales and introduces a rate function capturing directional differences.

Findings

Patterns appear on infinitely many time scales after rescaling

A functional law of iterated logarithm is established

A rate function differentiates one-sided and two-sided behaviors

Abstract

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.