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From random matrices to quasi-periodic Jacobi matrices via orthogonal polynomials. (English) Zbl 1118.15023

The author discusses asymptotic formulas for ordinary polynomials orthogonal with respect to weights whose support is a union of \(q\) disjoint intervals, presents asymptotics for orthogonal polynomials with respect to varying weights, then introduces quasi-periodic Jacobi matrices associated with the both asymptotics and discusses links between the matrices, and gives a collection of facts on asymptotic eigenvalue distributions of random matrices, that can be written in the terms of the above Jacobi matrices.

MSC:

15B52 Random matrices (algebraic aspects)
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
58J53 Isospectrality
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
15A18 Eigenvalues, singular values, and eigenvectors
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