Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension
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Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension Martin Prigent1 · Matthew I. Roberts1 Received: 30 November 2019 / Revised: 3 February 2020 © The Author(s) 2020
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 εn such that nεn → ∞. This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable. Keywords Random walk · Dynamical sensitivity · Exceptional times · Noise sensitivity · Hausdorff dimension Mathematics Subject Classification 60G50 · 82C41 · 28A78
1 Introduction and results Consider two simple symmetric random walks in one dimension. The first, at each step independently, jumps upwards with probability 1/2 or downwards with probability 1/2. The second begins facing upwards and, at each step independently, decides to take a step in the direction it is facing with probability 1/2; or switches direction and takes a step the other way with probability 1/2.
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Matthew I. Roberts [email protected] Martin Prigent [email protected]
1
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
123
M. Prigent, M. I. Roberts
We call the first of these two random walks the compass random walk, as it has an in-built sense of direction, and the second the switch random walk, as it only decides whether or not to switch directions. These two random walks have exactly the same distribution—they are simple symmetric random walks—although, as we will see when we define them rigorously, they are different functions of the underlying randomness. This means that when we talk about noise sensitivity or dynamical sensitivity of the two walks, they may (and do) have very different properties. We now define carefully the objects of interest. Let X 1 , X 2 , . . . be independent random variables satisfying P(X i = 1) = P(X i = −1) = 1/2 for each i ∈ N. Define, for each n ≥ 0, Yn =
n
Xj
j=1
and Zn =
n k
Xj
k=1 j=1
where we take the empty sum to be zero, so Y0 = Z 0 = 0. We call Y = (Yn , n ≥ 0) the compass random walk, and Z = (Z n , n ≥ 0) the switch random walk. We can think of Y = Y (X ) and Z = Z (X ) as functions of the sequence of random variables X = (X 1 , X 2 , . . .). It is easy to see that, although they are different functions, the two walks Y and Z have the same distribution. Indeed, the written descriptions at the beginning of thi
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