Bulk Universality and Related Properties of Hermitian Matrix Models

Abstract

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

References in corpus (4)

• Random Matrix Theories in Quantum Physics: Common Concepts
• 2D Gravity and Random Matrices
• Universality of Random-Matrix Results for non-Gaussian Ensembles
• A Note on the Eigenvalue Density of Random Matrices

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