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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 independen...
Comment: 6 pages, two figures .Presented at the International Symposium on Quantum Fluids and Solids QFS2006 (Kyoto, Japan)
Comment: Talk delivered by 3 pages, S.Chatterjee at at 17th DAE-BRNS High Energy Physics Symposium (HEP06), Kharagpur IIT, India, 11-16 Dec 2006
We propose a fairly simple and natural extension of Stollmann's lemma to correlated random variables. This extension allows (just as the original Stollmann's lemma does) to obtain Wegner-type estimates even in some problems of spectral analysis of random operators where the Wegner's lemma is inapplicable (e.g. for multi-particle Hamiltonians).
Comment: An invited talk at an International Symposium QBIC 2007
Expository paper on the relations between perturbation theory of pseudo-differential operators, finiteness theorems and deformations of Lagrangian varieties.
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