Accuracy of the Semicircle Approximation for the Density of Eigenvalues of Random Matrices

Certain statistical properties of the energy levels of complex physical systems have been found to coincide with those for distributions of eigenvalues derived from ensembles of random matrices. However, if ensembles of random matrices give a fair representation for the Hamiltonian of a complex physical system, the density of the characteristic values at the lower end of the spectrum should show some similarity with the exponential dependence found in nuclear spectra. The limiting distribution of the density for very high-dimensional random matrices is a semicircle, i.e., concave from below if plotted against the characteristic value which represents, in this case, the energy. Hence the deviations from the limiting distribution are investigated and it is shown that there is a region, at the very lowest part of the spectrum, where the density is convex from below, similar to an exponential function. The region of convexity is called the tail of the distribution. It is shown, however, that the avergae number of roots in the tail is very small, of the order of 1. It is concluded that those ensembles of random matrices which have been studied up to now, do not give a fair representation of Hamiltonians of complex systems.