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Strongly non-Gaussian ensembles of large random matrices possessing unitarysymmetry and logarithmic level repulsion are studied both in presence andabsence of hard edge in their energy spectra. Employing a theory of polynomialsorthogonal with respect to exponential weights we calculate with asymptoticaccuracy the two-point kernel over all distance scale, and show that in thelimit of large dimensions of random matrices the properly rescaled localeigenvalue correlations are independent of level confinement while globalsmoothed connected correlations depend on confinement potential only throughthe endpoints of spectrum. We also obtain exact expressions for density oflevels, one- and two-point Green's functions, and prove that new universallocal relationship exists for suitably normalized and rescaled connectedtwo-point Green's function. Connection between structure of Szeg\"o functionentering strong polynomial asymptotics and mean-field equation is traced.Comment: 12 pages (latex), to appear in Physical Review

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