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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 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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