Universality of the correlations between eigenvalues of large random matrices

The distribution of eigenvalues of random matrices appears in a number of physical situations, and it has been noticed that the resulting properties are universal, i.e. independent of specific details. Standard examples are provided by the universality of the conductance fluctuations from sample to sample in mesoscopic electronic systems, and by the spectrum of energy levels of a non-integrable classical hamiltonian (the so-called quantum chaos). The correlations between eigenvalues, measured on the appropriate scale, are described in all those cases by simple gaussian statistics. Similarly numerical experiments have revealed the universality of these correlations with respect to the probability measure of the random matrices. A simple renormalization group argument leads to a direct understanding of this universality; it is a consequence of the attractive nature of a gaussian fixed point. Detailed calculations of these correlations are given for a general probability distribution (in which the logarithm of the probability is the trace of a polynomial of the matrix); the universality is shown to follow from an explicit asymptotic form of the orthogonal polynomials with respect to a non-gaussian measure. In addition it is found that the connected correlations, when suitably smoothed, exhibit, even when the eigenvalues are not in the scaling region, a higher level of universality than the density of states.