If I am doing a random walk I can expect to be sqrt(n) steps away from the origin after taking n steps.
Is there anything I can expect about the *direction* though? I would think net direction would not drift much after a while. Max heading deviation = arctan(1/sqrt(n)) ...🤔
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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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That depends on how far you've drifted from the origin. If you're a long distance r from the origin, then you'll change your angle by ~ sqrt(t)/r during an interval t. It takes a time ~r^2 before your heading is changed completely.
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Do you just randomly know this shit off the top of your head (impressive) or figured it out on the fly (double impressive)?
Or is it a routine computation in solid state physics?
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