Theory of random matrices with strong level confinement: Orthogonal polynomial approach

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Abstract
Strongly non-Gaussian ensembles of large random matrices possessing unitary symmetry and logarithmic level repulsion are studied both in the presence and the absence of a hard edge in their energy spectra. Employing a theory of polynomials orthogonal with respect to exponential weights we calculate with an asymptotic accuracy the two-point kernel over all distance scale, and show that in the limit of large dimensions of random matrices the properly rescaled local eigenvalue correlations are independent of level confinement while global smoothed connected correlations depend on confinement potential only through the end points of the spectrum. We also obtain the exact expressions for density of levels, one- and two-point Green’s functions, and prove that a universal local relationship exists for the suitably normalized and rescaled connected two-point Green’s function. The connection between the structure of the Szegö function entering strong polynomial asymptotics and mean-field equation is traced. © 1996 The American Physical Society.

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