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The following is a subproblem of research project I'm working on, i.e. not a homework. Let's suppose you have a bounded 2d plane and n distinct probes that do random-walk in that plane. The world is closed in a sense that a probe going outside the border ends up being on the opposite side, e.g. a probe going too far east winds up showing up from the west.

Let's suppose you have a stationary circle of radius R in the plane. How long will it take before each of n probes pass through the circle at least once?

Thanks,

Monte