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Spectral and probabilistic aspects of matrix models. (English) Zbl 0844.15009

Boutet de Monvel, Anne (ed.) et al., Algebraic and geometric methods in mathematical physics. Proceedings of the 1st Ukrainian-French-Romanian summer school, Kaciveli, Ukraine, September 1-14, 1993. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 19, 207-242 (1996).
Summary: The paper deals with the eigenvalue statistics of \(n \times n\) random Hermitian matrices as \(n \to \infty\). We consider a certain class of unitary invariant matrix probability distributions which have been actively studied in recent years in the quantum field theory (QFT). These ensembles are natural extensions of the archetype Gaussian ensemble well known and widely studied in the field called random matrix theory (RMT) and having applications in a number of areas of physics and mathematics.
Our goal is to analyze the QFT motivated matrix ensembles from the point of view of the RMT. We consider the normalized counting functions of matrix eigenvalues (NCF), discuss the RMT content of various physical results (limiting form of the NCF, the eigenvalue spacing distribution, etc.), present rigorous versions and extensions of some of them and other rigorous results, and discuss open mathematical problems, conjectures, and links with other areas.
For the entire collection see [Zbl 0833.00031].

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15B52 Random matrices (algebraic aspects)
15A90 Applications of matrix theory to physics (MSC2000)
60E05 Probability distributions: general theory
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