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Abstract: We give a new proof of universality properties in the bulk of spectrum of thehermitian matrix models, assuming that the potential that determines the modelis globally $C^{2}$ and locally $C^{3}$ function (see Theorem ef{t:U.t1}).The proof as our previous proof in \cite{Pa-Sh:97} is based on the orthogonalpolynomial techniques but does not use asymptotics of orthogonal polynomials.Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linearintegro-differential equation that follows from the determinant formulas forthe correlation functions of the model. We also give a simplified andstrengthened version of paper \cite{BPS:95} on the existence and properties ofthe limiting Normalized Counting Measure of eigenvalues. We use these resultsin the proof of universality and we believe that they are of independentinterest.



Autor: L.Pastur, M.Shcherbina

Fuente: https://arxiv.org/







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