Suppose $V$ is a $N\times n$ matrix the columns of which are independently distributed uniformly on $\mathbf S^{N-1}$ the surface of the unit sphere in $\mathbf R^N$. I conjecture that $V^TV$ approaches the identity matrix in norm (say, norms that are equivalent to the Frobenius norm), as $N\to\infty$ in expectation or even almost surely. I can prove it for $n=2$. I need one for arbitrarily given $n$. In general, what is the joint distribution of $V^TV$ for given finite $n$ and $N$?

Is the conjecture correct? Do we need the theory of random matrix to obtain the answers?