Let $x_i^{(n)}$ (with $i=1,\ldots,d$ and $n=1,\ldots,N$) be $d\times N$ i.i.d variables, normally distributed with zero mean and unit variance. What is the mean euclidean distance between the vector $(x_1^{(1)},x_2^{(1)}, \ldots, x_d^{(1)})$ and its nearest neighbor among the $N-1$ vectors $(x_1^{(k)},x_2^{(k)}, \ldots, x_d^{(k)})$ for $k=2, \ldots, N$?
Edit: I'm particularly interested in knowing if the nearest-neighbor distance grows linearly with $d$.