Scaling laws of the additive random-matrix model

Abstract

The scaling behavior of the additive random-matrix model H=(${\mathit{H}}^0$+λV)/(1+${\lambda }^2$ ${)}^{1/2}$ has been investigated. This model, capable of producing transitions between different degrees of level repulsion for the eigenvalues, has been analyzed for the case of the transition from the Poisson distribution exp(-S) of the distances between adjacent eigenvalues to the Wigner distribution (πS/2)exp(-${\mathit{S}}^2$π/4). We propose expressions for the level-spacing distribution and the distribution of the eigenvector components and demonstrate their reliability by means of numerical computations. For the eigenvalues, the transition proceeds at a rate that scales as the square root of the matrix dimension. For the eigenvalues, the rate is independent of this dimension.