Dr. Nestor Parolya Assistant professor TUDelft/EEMCS/DIAM van Mourik Broekmanweg 6 2628 XE Delft Phone: 0152782515 Office: 28.1.E200 Email: n.parolya@tudelft.nl 
What is the true relationship between the eigenvalues of the highdimensional sample covariance matrix with limiting spectral distribution F and its population counterpart H if p/n (dimension/sample size) tends to a constant c>0? Integral fixed point equation in terms of the corresponding Stieltjes transforms!
The solution is the MarchenkoPastur law.

Toy example: let the true covariance matrix is equal to identity matrix. How good the sample covariance matrix S_n estimates the true one in terms of its eigenvalues if dimension p=500 and the sample size n=1000? The picture says it all.

"Why should the same equation describe both the structure of an atomic nucleus and a sequence at the heart of number theory? And what do random matrices have to do with either of those realms? In recent years, the plot has thickened further, as random matrices have turned up in other unlikely places, such as games of solitaire, onedimensional gases and chaotic quantum systems. Is it all just a cosmic coincidence, or is there something going on behind the scenes?"  Brian Hayes, The Spectrum Of Riemannium, American Scientist (2003)