That is, the behavior of L 1 is qualitatively different depending on whether l 1 is greater or less than 1 + 1/γ. This is a phase-change phenomenon, and we will define as the BBP threshold. This is an asymptotic result, showing that as the data size goes to infinity, the transition of the behavior, as l 1 varies, becomes arbitrarily sharp. The result, as stated above, is proved in [16] for data where the matrix entries are complex numbers, and statement (2) of the conjecture is proved in [17], which demonstrates that the behavior is qualitatively different according to whether l 1 is greater or less than 1 + 1/ γ. There seems little doubt as to the truth of statement (1) above. It has been shown (D. Paul, Asymptotic behavior of the leading sample eigenvalues for a spiked covariance model, http://anson.ucdavis.edu/~debashis/techrep/eigenlimit.pdf) that, under the assumptions of statement (1) above, the lead eigenvector of the sample covariance is asymptotically uncorrelated with the lead eigenvector of the theoretical covariance, but we believe that the question of the distribution of the leading eigenvalue is still open.
