Random nigger walk

Suppose a gang of an arbitrary number of niggers. The first nigger can move in only one dimension, and each subsequent nigger can move in one dimension greater than the last (ie, nigger No. 12 can move in twelve dimensions). The niggers live in a city of arbitrary dimensions that expand forever in all directions as an infinite ghetto. At point p, the niggers participate in a walk-by shooting. The police show up and the niggers scatter. Each nigger, upon reaching an intersection, picks a random direction to run (the nigger elects to turn back with equal probability) from all the dimensions he can travel. If a nigger returns to p, he is arrested. The niggers run forever. What are the probabilities that any given number of niggers will be caught?

This is actually a complete pain in the ass to try to solve. Two hours in and the only one approximated is d=3. The slightest change in the trapezoid integration blows all the other probabilities up. It takes and enormous amount of work to solve it by hand, which doesn't even really help tell me where it's going wrong.

I'm tired. This is annoying. I don't care anymore. I give up. The hyperniggers go free. You win, >>1-sama. I'm going to bed.

Fuck Bessel.