Before revision:  We show that this critical value is universal, independent of the distribution of the returns. Beyond this critical value, the variance of the portfolio vanishesfor any portfolio weight vector constructed as a linear combination of the eigenvectors from the null space of the covariance matrix, but these linear combinations are not legitimate solutions of the optimization problem, as they are infinitely sensitive to any change in the input parameters, in particular they will wildly fluctuate from sample to sample.
After revision:  At r=2 the out of sample estimator for the portfolio variance stays finite and the estimated in-sample variance vanishes. We have performed numerical simulations to support the analytic results and found perfect agreement for N/T<2. Numerical experiments on finite size samples of symmetrically distributed returns show that above this critical point the probability of finding solutions with zero in-sample variance increases rapidly with increasing N, becoming one in the large N limit. However, these are not legitimate solutions of the optimization problem, as they are infinitely sensitive to any change in the input parameters, in particular they will wildly fluctuate from sample to sample.