On eigenvalue distribution in some ensembles of large random matrices

In the paper the differential equation obtained by V. A. Marchenko and L. A. Pastur [1] is examined which describes spectral distribution in some ensembles of large random matrices. The solution of this equation is explicitly found, as well as the rule proposed in [1] for finding out intervals on real axis complement to spectrum is proven. The methods of V. A. Marchenko and L. A. Pastur are applied in the neural networks theory for studying evolution of spectrum of interneuron connection matrix describing REM sleep. Asymptotic behavior of spectrum is investigated; it is proven to differ qualitatively in cases where a parameter $\alpha$ corresponding to memory loading with memorizing patterns is less than some critical value $\alpha_c$, and where $\alpha>\alpha_c$. From viewpoint of associative memory in neural networks, all the patterns are memorized as a result of sleeping in the first case, and are not in the second one.