Large Random Matrices: Eigenvalue Distribution

A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is universal, is recovered and the three and four-point functions are given explicitly. One observes that higher order correlation functions are linear combinations of universal functions with coefficients depending on an increasing number of parameters of the matrix distribution. 11/93, for Nuclear Physics B *Laboratoire de la Direction des Sciences de la Matière du Commissariat à l’Energie Atomique In a recent article [1], Brézin and Zee have calculated explicitly correlation functions of eigenvalues of a class of stochastic hermitian matrices of large size N . They have found that some statistical properties of the eigenvalues are universal in the large N limit, and can thus be obtained from the correlation functions of the gaussian model. More precisely, they have discovered that the two-point correlation function, after smoothing of the short scale oscillations, is universal while all other correlations vanish at the same order. This property can be related to a renormalization group analysis [2,3] which has shown that the gaussian model is a stable fixed point in the large N limit. Their analysis is based on the, by now standard, method of orthogonal polynomials. An essential ingredient in the final answer is a proposed ansatz for an asymptotic form of the orthogonal polynomials Pn in the limit N → ∞ and N − n finite. In ref. [1] the ansatz is verified in the case of even integrands, and only up to an unknown function. Here, we propose a direct proof of the ansatz, using a saddle point method, which does not depend on the parity of the integrand, and which allows to determine the previously unknown function. Moreover, using a completely different approach [4], we present a recursive method to evaluate all smoothed correlation functions at leading order. We give the three and four-point functions explicitly. 1. Correlation functions of eigenvalues Let us first explain the problem and recall the method used in ref. [1] to explicitly evaluate the eigenvalue correlation functions. We consider N ×N hermitian matrices M with a probability distribution of the form: P(M) = 1 Z e −NtrV (M) , (1.1) where V (M) is a polynomial, and Z the normalization (i.e. the partition function). We want to derive the asymptotic form for N large of various eigenvalue correlation functions. All can be derived from the correlation functions of the operator O(λ): O(λ) = 1 N tr δ(λ−M) = 1 N N ∑