[hemmer]

There is some more information and references here: http://mathworld.wolfram.com/PolyasRandomWalkConstants.html

[Ntrails]

If you find it GP, I'd love to see where the integral constructed comes from, since that's the clever part rather than the evaluation.

[emfree]

Here's a reference I found for one way to do it: http://www.math.nus.edu.sg/~matsr/ProbII/Lec6.pdf (Theorem 2.1). You define the Green's function G(x, y) = \sum_n Pr_x(S_n=y), where x and y are 3-vectors and Pr_x(S_n=y) is the probability that an n-step random walk starting at x ends up at y. If you have an infinite random walk starting at 0, then G(0, 0) is the expected number of times that the walk returns to 0. That's what the mathworld link calls u(3). You can use Fourier inversion to compute G(0, 0) -- the link gives the gnarly details. It's pretty cool.

[Ntrails]

You're a scholar and a gentleman, merci buckets

[ccvannorman]

The probability of returning to the origin follows a very smooth logarithmic curve for dimensions 3 - 8 [copied values from the mathworld link]

image: http://imgur.com/CL8MXej

[sdoering]

Thanks a lot. Had to save this for further reference the moment I saw it.