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A RNG problem. I'm trying to uniformly sample paths on an SW --> NE grid, of size NS > EW, via random walk. So I generate a random number between 0 and 1 and go east if r < EW/(NS+EW)... but this seems to create a bias towards denser sampling of latter half of path
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Then you generate a strong SE bias because and your sampling has an off-diagonal bias. That was what I did initially. Now I've fixed the diagonal bias, but not the terminal bias.
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The walk is not truly random because as you progress, the area where you have degrees of freedom decreases. So there will always be more paths in the NE corner as the walk progresses unless you allow motions that can go W and S.
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The basic issue here is that time is an asymmetry here and you're using up degrees of freedom as you go unequally, and due to discreteness, you'll run out of one or the other first. In the infinitesimal limit, I think you'd recover time symmetry. Or something like that.
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