Are we in one dimension here? In 1D, if you are a distance L away from the origin, then you will probably stay on the same side of the origin for a time ~L^2. Also, if you start at the origin, then the probability of making it to L without first returning to the origin is ~1/L.
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That’s a useful reduction. I guess the 2d equivalent would be to compute expected dwell time in a tolerance range, +/- theta from current heading
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Highly, um, path dependent on the bearing of the first few steps.
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Or more poetically: compared to origin your direction gets more and more narrowly defined. Compared to yourself a few steps ago, you are just as aimless as ever.
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I knew it! Among the elementary functions, your absolute *favorite* is the tangent!
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