Universal behavior of correlations between eigenvalues of random matrices.

The universal connected correlations proposed recently between eigenvalues of unitary random matrices is examined numerically. We perform an ensemble average by the Monte Carlo sampling. Although density of eigenvalues and a bare correlation of the eigenvalues are not universal, the connected correlation shows a universal behavior after smoothing. Typeset using REVTEX 1 Brézin and Zee [1–3] have discovered recently the universal behavior of the eigenvalue correlation of random matrices. Let φ be N ×N hermitian matrices and let ith eigenvalue of φ be denoted by λi, μi or νi. We consider the probability distribution with a weight P (φ) ∝ exp(−N TrV (φ)) (1) where V (φ) is an even polynomial of φ. The density of the eigenvalues ρ(λ) and the correlation of the eigenvalues ρ(μ, ν) are defined as ρ(λ) = 〈 1 N N