Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension

TL;DR

This paper introduces a dynamical model of a one-dimensional symmetric random walk, demonstrating the existence of exceptional times with diverging behavior, and explores the noise sensitivity of the walk's positivity event.

Contribution

It establishes the existence and properties of exceptional times in a dynamical setting, contrasting with classical results, and analyzes the noise sensitivity of the walk's positivity event.

Findings

Exceptional times with diverging behavior exist almost surely.

The set of exceptional times has Hausdorff dimension 1/2.

The positivity event is maximally noise sensitive.

Abstract

We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension $1/2$ almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after $n$ steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence $\varepsilon_n$ such that $n\varepsilon_n\to\infty$. This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.