As you let angle of slice go to zero, it will become a straight line, so yes?
Quick math question: if you constrain a random walk so that the system does not respond to impulses in all but a tiny pizza slice, the distance from origin should grow in the slice direction linearly in number of steps, right? n instead of sqrt(n)?
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So unconstrained is symmetric random walk with steps +1,-1, probability 1/2 each direction, how exactly do you constrain it? The pizza is in space/time? Can you provide more details?
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I assumed: at every epoch, choose a random angle a uniformly at random from [0, 2pi). If a < some small angle d, then take a step of length 1 in the direction of a. Otherwise, choose another angle. Repeat for n steps. How does distance from origin scale with n as d -> 0.
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At least, a symmetric simple random walk that cannot go through 0 [i.e. barrier at 0], also grows with rate sqrt n
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