Asymptotic properties of polynomials orthogonal with respect to varying weights, and related topics of spectral theory
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I. Egorova and L. Pastur
Translated by: the authors - St. Petersburg Math. J. 25 (2014), 223-240
- DOI: https://doi.org/10.1090/S1061-0022-2014-01287-3
- Published electronically: March 12, 2014
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Abstract:
A detailed description is given of links (outlined in a paper by the second author) between asymptotic formulas for polynomials orthogonal with respect to varying exponential weights and finite band Jacobi operators.References
- N. I. Ahiezer, Orthogonal polynomials on several intervals, Soviet Math. Dokl. 1 (1960), 989–992. MR 0110916
- N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
- N. I. Ahiezer and Ju. Ja. Tomčuk, On the theory of orthogonal polynomials over several intervals, Dokl. Akad. Nauk SSSR 138 (1961), 743–746 (Russian). MR 0131005
- A. I. Aptekarev, Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains, Mat. Sb. (N.S.) 125(167) (1984), no. 2, 231–258 (Russian). MR 764479
- —, Matrix Riemann–Hilbert analysis for the case of higher genus — asymptotics of polynomials orthogonal on a system of intervals, KIAM Preprint, Moscow, 28 (2008), 1–23. (Russian)
- D. Bessis, C. Itzykson, and J. B. Zuber, Quantum field theory techniques in graphical enumeration, Adv. in Appl. Math. 1 (1980), no. 2, 109–157. MR 603127, DOI 10.1016/0196-8858(80)90008-1
- Pavel Bleher and Alexander Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. (2) 150 (1999), no. 1, 185–266. MR 1715324, DOI 10.2307/121101
- Pavel Bleher and Alexander Its (eds.), Random matrix models and their applications, Mathematical Sciences Research Institute Publications, vol. 40, Cambridge University Press, Cambridge, 2001. MR 1842779
- G. Bonnet, F. David, and B. Eynard, Breakdown of universality in multi-cut matrix models, J. Phys. A 33 (2000), no. 38, 6739–6768. MR 1790279, DOI 10.1088/0305-4470/33/38/307
- Édouard Brézin, Vladimir Kazakov, Didina Serban, Paul Wiegmann, and Anton Zabrodin (eds.), Applications of random matrices in physics, NATO Science Series II: Mathematics, Physics and Chemistry, vol. 221, Springer, Dordrecht, 2006. MR 2238825, DOI 10.1007/1-4020-4531-X
- V. Buslaev and L. Pastur, A class of the multi-interval eigenvalue distributions of matrix models and related structures, Asymptotic combinatorics with application to mathematical physics (St. Petersburg, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 77, Kluwer Acad. Publ., Dordrecht, 2002, pp. 51–70. MR 1999355
- V. S. Buyarov and E. A. Rakhmanov, On families of measures that are balanced in the external field on the real axis, Mat. Sb. 190 (1999), no. 6, 11–22 (Russian, with Russian summary); English transl., Sb. Math. 190 (1999), no. 5-6, 791–802. MR 1719585, DOI 10.1070/SM1999v190n06ABEH000407
- P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1677884
- Percy A. Deift, Alexander R. Its, and Xin Zhou, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math. (2) 146 (1997), no. 1, 149–235. MR 1469319, DOI 10.2307/2951834
- P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, 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 (1999), no. 11, 1335–1425. MR 1702716, DOI 10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1
- P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), no. 12, 1491–1552. MR 1711036, DOI 10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.3.CO;2-R
- Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745
- Roberto Fernández, Jürg Fröhlich, and Alan D. Sokal, Random walks, critical phenomena, and triviality in quantum field theory, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. MR 1219313, DOI 10.1007/978-3-662-02866-7
- A. S. Fokas, A. R. It⋅s, and A. V. Kitaev, Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), no. 2, 313–344. MR 1137067
- A. S. Fokas, A. R. It⋅s, and A. V. Kitaev, The isomonodromy approach to matrix models in $2$D quantum gravity, Comm. Math. Phys. 147 (1992), no. 2, 395–430. MR 1174420
- P. J. Forrester, Log-gases and random matrices, London Mathematical Society Monographs Series, vol. 34, Princeton University Press, Princeton, NJ, 2010. MR 2641363, DOI 10.1515/9781400835416
- A. B. J. Kuijlaars and K. T-R McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Comm. Pure Appl. Math. 53 (2000), no. 6, 736–785. MR 1744002, DOI 10.1002/(SICI)1097-0312(200006)53:6<736::AID-CPA2>3.0.CO;2-5
- K. T.-R. McLaughlin and P. D. Miller, The $\overline {\partial }$ steepest descent method for orthogonal polynomials on the real line with varying weights, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 075, 66. MR 2439564, DOI 10.1093/imrn/rnn075
- L. A. Pastur, Spectral and probabilistic aspects of matrix models, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993) Math. Phys. Stud., vol. 19, Kluwer Acad. Publ., Dordrecht, 1996, pp. 207–242. MR 1385683
- L. Pastur, From random matrices to quasi-periodic Jacobi matrices via orthogonal polynomials, J. Approx. Theory 139 (2006), no. 1-2, 269–292. MR 2220042, DOI 10.1016/j.jat.2005.09.006
- Leonid Pastur and Alexander Figotin, Spectra of random and almost-periodic operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297, Springer-Verlag, Berlin, 1992. MR 1223779, DOI 10.1007/978-3-642-74346-7
- Leonid Pastur and Mariya Shcherbina, Eigenvalue distribution of large random matrices, Mathematical Surveys and Monographs, vol. 171, American Mathematical Society, Providence, RI, 2011. MR 2808038, DOI 10.1090/surv/171
- Franz Peherstorfer and Peter Yuditskii, Asymptotic behavior of polynomials orthonormal on a homogeneous set, J. Anal. Math. 89 (2003), 113–154. MR 1981915, DOI 10.1007/BF02893078
- Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778, DOI 10.1007/978-3-662-03329-6
- Barry Simon, Szegő’s theorem and its descendants, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011. Spectral theory for $L^2$ perturbations of orthogonal polynomials. MR 2743058
- Mikhail Sodin and Peter Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. 7 (1997), no. 3, 387–435. MR 1674798, DOI 10.1007/BF02921627
- Herbert Stahl and Vilmos Totik, General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. MR 1163828, DOI 10.1017/CBO9780511759420
- S. P. Suetin, On trace formulas for a class of Jacobi operators, Mat. Sb. 198 (2007), no. 6, 107–138 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 5-6, 857–885. MR 2355367, DOI 10.1070/SM2007v198n06ABEH003864
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 1711536, DOI 10.1090/surv/072
- Vilmos Totik, Weighted approximation with varying weight, Lecture Notes in Mathematics, vol. 1569, Springer-Verlag, Berlin, 1994. MR 1290789, DOI 10.1007/BFb0076133
- Harold Widom, Extremal polynomials associated with a system of curves in the complex plane, Advances in Math. 3 (1969), 127–232. MR 239059, DOI 10.1016/0001-8708(69)90005-X
Bibliographic Information
- I. Egorova
- Affiliation: B. Verkin Institute for Low Temperature Physics, Lenin Avenue 47, Kharkiv 61103, Ukraine
- MR Author ID: 213624
- Email: iraegorova@gmail.com
- L. Pastur
- Affiliation: B. Verkin Institute for Low Temperature Physics, Lenin Avenue 47, Kharkiv 61103, Ukraine
- Email: pastur2001@yahoo.com
- Received by editor(s): November 15, 2012
- Published electronically: March 12, 2014
- © Copyright 2014 American Mathematical Society
- Journal: St. Petersburg Math. J. 25 (2014), 223-240
- MSC (2010): Primary 37K40, 35Q53; Secondary 37K45, 35Q15
- DOI: https://doi.org/10.1090/S1061-0022-2014-01287-3
- MathSciNet review: 3114851