What is the expectation of the average magnitude of correlation in a uniform multidimensional random set?

Question

When creating random samples independly for each variable dimension we should have no correlation. But if you have many dimensions d, and calculate the pairwise correlation cij, you will find that sometimes the magnitude of correlation (using the linear correlation factor) is quite significant, just by chance! I wonder how large is e.g. the expected average or maximum correlation? Example: You create a 3D set of independant uniform variates, so we can calculate 2D correlation between x1,2, x1,3 and x2,3, like c12=-0.1, c13=0, c23=+0.1 (or whatever), so the maximum correlation magnitude is 0.1 and the average magnitude is 0.066. How do these two calculated values depend on point count N and number of variables d? The background is that I want to compare low-discrepancy samples (LDS) again random samples on correlation. Also LDS should have no correlations, but as most LDS methods are not perfect, there is of course some correlation. Also a result for normal variates (instead of uniform) would be helpful. Also the expectation of c*c (if this is easier to calculate).