Distribution of eigenvalues of certain matrix ensembles

We investigate spectral properties of ensembles of N×N random matrices M defined by their probability distribution P(M)=exp[- Tr V(M)] with a weekly confinement potential V(M) for which the moment problem ${\mathrm{\mu }}_{\mathrm{n}}$=∫${\mathrm{x}}^{\mathrm{n}}$exp[-V(x)]dx is indeterminated. The characteristic property of these ensembles is that the mean density of eigenvalues tends with increasing matrix dimension to be a continuous function contrary to the usual strong confinement cases, where it grows indefinitely when N→∞. We demonstrate that the standard asymptotic formulas for correlation functions are not applicable for weakly confinement ensembles and their asymptotic distribution of eigenvalues can deviate from the classical ones. The model with V(x)=ln ${}_{}{}^2{}_{}{}^{}$(|x|)/Β is considered in detail. It is shown that when Β→∞ the unfolded eigenvalue distribution tends to a limit which is different from any standard random matrix ensembles, but which is the same for all three symmetry classes: unitary, orthogonal, and symplectic.