I'm having a hard time believing that a three dimensional random walk won't eventually lead back to the origin (infinite times). Just as a monkey will type any finite length string infinite times, any trip back to the origin is just a string of directions. I don't see how adding dimensions changes anything. Even if our monkey is at the edge of the observable universe, he has a nonzero chance of traveling the roughly 10^60 planck lengths forward in dimension 1, 10^60 planck lengths forward in dimension 2, 10^60 planck lengths backwards in dimension 3. In fact our monkey has a nonzero chance of traveling back taking the exact same route with all the whoopty swirls that led him out there to begin with. I didn't realize you were alive in the year infinity B.C. Otherwise your program hasn't gone through infinite iterations yet.