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.


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
Full Text: DOI arXiv Link


[1] Achiezer, N., Uber einige funktionen welche in zwei gegebenen intervallen am wenigsten von null abweichen, Izv. akad. nauk SSSR, 3, 309-344, (1933) · Zbl 0007.34106
[2] Akhiezer, N.I., Orthogonal polynomials on several intervals, Soviet math. dokl., 1, 989-992, (1960) · Zbl 0101.29205
[3] Akhiezer, N., Classical moment problem, (1965), Haffner New York · Zbl 0135.33803
[4] Akhiezer, N.I., Elements of the theory of elliptic functions, (1990), AMS Providence · Zbl 0694.33001
[5] Akhiezer, N.; Tomchuk, Yu., On the theory of orthogonal polynomials over several intervals, Dokl. akad. nauk SSSR, 138, 743-745, (1961) · Zbl 0109.29601
[6] Albeverio, S.; Pastur, L.; Shcherbina, M., On asymptotic properties of certain orthogonal polynomials, Math. phys. anal. geom., 4, 263-277, (1997) · Zbl 0897.33007
[7] Aptekarev, A.I., Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains, Math. USSR-sb., 53, 233-260, (1986) · Zbl 0608.42016
[8] Bleher, P.; Its, A., Semiclassical asymptotics of orthogonal polynomials, riemann – hilbert problem, and universality in the matrix model, Ann. math., 150, 185-266, (1999) · Zbl 0956.42014
[9] Bonnet, G.; David, F.; Eynard, B., Breakdown of universality in multi-cut matrix models, J. phys. A, 33, 6739-6768, (2000) · Zbl 0963.82021
[10] Boutet de Monvel, A.; Pastur, L.; Shcherbina, M., On the statistical mechanics approach to the random matrix theory: the integrated density of states, J. statist. phys., 79, 585-611, (1995) · Zbl 1081.82569
[11] Buslaev, V.; Pastur, L., A class of multi-interval eigenvalue distributions of matrix models and related structures, (), 52-70 · Zbl 1041.81024
[12] Buyarov, V.S.; Rakhmanov, E.A., Families of equilibrium measures in an external field on the real axis, Sb. math., 190, 791-802, (1999) · Zbl 0933.31002
[13] Cycon, H.L.; Froese, R.G.; Kirsch, W.; Simon, B., Schrödinger operators with application to quantum mechanics and global geometry, (1987), Springer Berlin · Zbl 0619.47005
[14] Deift, P.; Kriecherbauer, T.; McLaughlin, K., New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. approx. theory, 95, 388-475, (1998) · Zbl 0918.31001
[15] Deift, P.; Kriecherbauer, T.; McLaughlin, K.; Venakides, S.; Zhou, X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. pure appl. math., 52, 1335-1425, (1999) · Zbl 0944.42013
[16] DiFrancesco, P.; Ginsparg, P.; Zinn-Justin, J., 2D gravity and random matrices, Phys. rep., 254, 1-133, (1995)
[17] Dubrovin, B.A.; Krichever, I.M.; Novikov, S.P., Integrable systems. I, (), 173-280 · Zbl 0780.58019
[18] Guhr, T.; Mueller-Groeling, A.; Weidenmueller, H.A., Random matrix theories in quantum physics: common concepts, Phys. rept., 299, 189-425, (1998)
[19] Johansson, K., On fluctuations of eigenvalues of random Hermitian matrices, Duke math. J., 91, 151-204, (1998) · Zbl 1039.82504
[20] Katz, N.; Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy, (1999), AMS Providence · Zbl 0958.11004
[21] Kuijlaars, A.B.J.; McLaughlin, K., Generic behaviour of the density of states in random matrix theory and equilibrium problem of real analytic external field, Comm. pure appl. math., 53, 736-785, (2000) · Zbl 1022.31001
[22] Levitan, B., Inverse sturm – liouville problems, (1987), VNU Science Press Utrecht · Zbl 0749.34001
[23] Marchenko, V., Sturm – liouville operators and applications, (1986), Birkhauser Basel
[24] Marchenko, V.; Ostrovski, I., A characterization of the spectrum of the Hill operator, Math. USSR-sb., 25, 493-554, (1975)
[25] Mehta, L., Random matrices, (1991), Academic Press New York
[26] Pastur, L., Spectral and probabilistic aspects of matrix models, (), 207-242 · Zbl 0844.15009
[27] Pastur, L., Random matrices as paradigm, (), 216-266 · Zbl 1017.82023
[28] L. Pastur, Limiting laws for linear eigenvalue statistics of matrix models, to be published. · Zbl 1112.82022
[29] Pastur, L.; Figotin, A., Spectra of random and almost periodic operators, (1992), Springer Berlin · Zbl 0752.47002
[30] Pastur, L.; Shcherbina, M., Universality of the local eigenvalue statistics for a class of unitary invariant matrix ensembles, J. statist. phys., 86, 109-147, (1997) · Zbl 0916.15009
[31] Peherstorfer, F., On bernstein – szegö orthogonal polynomials on several intervals, II: orthogonal polynomials with periodic recurrence coefficients, J. approx. theory, 64, 123-161, (1991) · Zbl 0721.42017
[32] Peherstorfer, F., Elliptic orthogonal and extremal polynomials, Proc. London math. soc. (3), 70, 605-624, (1995) · Zbl 0833.42012
[33] Peherstorfer, F.; Yuditskii, P., Asymptotic behavior of polynomials orthonormal on a homogeneous set, J. anal. math., 89, 113-154, (2003) · Zbl 1032.42028
[34] Saff, E.B.; Totik, V., Logarithmic potentials with external fields, (1997), Springer New York · Zbl 0881.31001
[35] Sodin, M.; Yuditskii, P., Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. geom. anal., 7, 387-435, (1997) · Zbl 1041.47502
[36] Simon, B., Orthogonal polynomials on the unit circle, parts 1 and 2, (2005), AMS Providence
[37] Szegö, G., Orthogonal polynomials, (1975), AMS Providence · JFM 65.0278.03
[38] Teschl, G., Jacobi operators and completely integrable systems, (1999), AMS Providence
[39] V. Totik, Weighted Approximation with Varying Weight, Lecture Notes in Mathematics, vol. 1569, Springer, Berlin, 1994. · Zbl 0808.41001
[40] Widom, H., Extremal polynomials associated with a system of curves in the complex plane, Adv. math., 3, 127-232, (1969) · Zbl 0183.07503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.