Higher Dimensional Random Walks

Question

I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in an arbitrary direction (uniformely on the unit sphere $S^1$), the expected distance after $n$ steps from the starting point is approximated by $\sqrt{n\pi}/2$.

(source: https://pdfs.semanticscholar.org/6685/1166d588821456477f2007a37bf0428a2cf2.pdf).

I was wondering if there was a similar formula for higher dimensional random walks, which means:

Starting from the origin in $\mathbb{R}^d$, if i set $v_i = v_{i-1} + \mu$, where $\mu \in S^{d-1}$ picked at random at every step and $v_0 = 0$, what is the expected value of $|v_n|$? e.g. how distant is the point from the origin after having taken $n$ steps in random directions?

I don't need a precise formula, everything that just gives an idea on how large the expected distance is works just fine. (if you could add a reference to the answer as well it would be great! :D )

Thank you very much!

Answer 1

As described here:

$${\displaystyle P(r)={\frac {2r}{N}}e^{-r^{2}/N}}$$

Answer 2

The trend is presumably further away for higher dimensions. With endless paths in fewer than 3 dimensions the origin will be endlessly revisited. With 3 or more dimensions once enough space has been you and the origin you'll likely never return. Negatively curved space also counts against returning, 2D should be enough in this case for the distance to tend towards infinity.