Unitary Random-Matrix Ensemble with Governable Level Confinement

Abstract

A family of unitary $\alpha$-Ensembles of random matrices with governable confinement potential $V(x)|x{|}^{\alpha }$ is studied employing exact results of the theory of non-classical orthogonal polynomials. The density of levels, two-point kernel, locally rescaled two-level cluster function and smoothed connected correlations between the density of eigenvalues are calculated for strong ($\alpha >1$) and border ($\alpha =1$) level confinement. It is shown that the density of states is a smooth function for $\alpha >1$, and has a well-pronounced peak at the band center for $\alpha <=1$. The case of border level confinement associated with transition point $\alpha =1$ is reduced to the exactly solvable Pollaczek random-matrix ensemble. Unlike the density of states, all the two-point correlators remain (after proper rescaling) to be universal down to and including $\alpha =1$.

14 pages (revtex), 4 figures available upon request

References in corpus (8)

• Universal Spectral Correlations at the Mobility Edge
• A generalized ensemble of random matrices
• Universality of Brezin and Zee's Spectral Correlator
• Universality of Random-Matrix Results for non-Gaussian Ensembles
• Transitions In Spectral Statistics
• Normalization Sum Rule and Spontaneous Breaking of U(N) Invariance in Random Matrix Ensembles
• Non-Universality in Random Matrix Ensembles with Soft Level Confinement
• Gap Formation Probability of the $\alpha -$ Ensemble

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