Solving Random Walk Question in 2D Plane | Monte

[Monte_Carlo]

Hello Everyone,

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

 

[mfb]

If you can find the probability that a single probe passed through the circle after N steps you can construct the probability that all of them did. I would be surprised if there was an analytic solution, but simulations should work.