For the switch random walk, exceptional times of transience exist almost surely, the exceptional set has Hausdorff dimension 1/2, and the positivity event is maximally noise sensitive.
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Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension
For the switch random walk, exceptional times of transience exist almost surely, the exceptional set has Hausdorff dimension 1/2, and the positivity event is maximally noise sensitive.