Distribution of sum of $n$ random unit vectors in $d$-dimensional space

Question

I'm interested in the following (equivalent) problems, for $d$ around 100-1000, and $n$ fairly small (perhaps up to logarithmic in $d$?):

1. Given $n$ random unit vectors in $d$-dimensional space, what is the probability that their sum has norm at most $\alpha$?
2. How likely is it that an $n$-step random walk in $d$-dimensional space ends up within distance $\alpha$ of the origin?

I've found one paper (Equation 12) but the results are huge integrals filled with Bessel functions. Really, I only want the largest terms; for example, when $n=2$, the probability is $O(poly(d)\sqrt{1-\alpha^2}^d)$, and that's good enough (although if the $d$ polynomial varies in $n$, I'd like to know). I tried to bound or approximate the Bessel functions, but didn't get anywhere. A reasonable lower bound would also be fine.

Any ideas and/or references?