Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Asymptotic properties of polynomials orthogonal with respect to varying weights, and related topics of spectral theory


Authors: I. Egorova and L. Pastur
Translated by: the authors
Original publication: Algebra i Analiz, tom 25 (2013), nomer 2.
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
Published electronically: March 12, 2014
MathSciNet review: 3114851
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. N. I. Akhiezer, Orthogonal polynomials on several intervals, Dokl. Akad. Nauk SSSR 134 (1960), no. 1, 9-12; English transl., Soviet Math. Dokl. 1 (1960), 989-992. MR 0110916 (22:1784)
  • 2. -, The classical moment problem and some related questions in analysis, Hafner Publ. Co., New York, 1965. MR 0184042 (32:1518)
  • 3. N. I. Akhiezer and Yu. Tomchuk, On the theory of orthogonal polynomials over several intervals, Dokl. Akad. Nauk SSSR 138 (1961), no. 4, 743-745. (Russian) MR 0131005 (24:A859)
  • 4. A. I. Aptekarev, Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains, Math. Sb. (N.S.) 125 (1984), no. 2, 231-258; English. transl., Math. USSR-Sb. 53 (1986), 233-260. MR 764479 (86g:35166)
  • 5. -, 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)
  • 6. 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 (83j:81067)
  • 7. P. M. Bleher and A. R. Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. Math.(2) 150 (1999), no. 1, 185-266. MR 1715324 (2000k:42033)
  • 8. -, Random matrix models and their applications, Edited by P. Bleher and A. Its. Cambridge Univ. Press, Cambridge, 2001. MR 1842779 (2002a:82002)
  • 9. 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 (2001h:82042)
  • 10. E. Brézin, V. Kazakov, D. Serban, P. Wiegmann, and A. Zabrodin, Applications of random matrices in physics, NATO Sci. Ser.II Math. Phys. Chem., 221, Springer, Dordrecht, 2006. MR 2238825 (2007a:82001)
  • 11. V. S. Buslaev and L. Pastur, A class of multi-interval eigenvalue distributions of matrix models and related structures, Asymptotic combinatorics with applications to mathematical physics (St. Petersburg, 2001), 51-70, NATO Sci. Ser. II Math. Phys. Chem., 77, Kluwer, Acad. Publ., Dordrecht, 2002. MR 1999355 (2004h:82044)
  • 12. 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; English transl., Sb. Math. 190 (1999), no. 5-6, 791-802. MR 1719585 (2001b:31002)
  • 13. P. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Math., 3, New York Univ., Courant Inast. Math. Sci., Amer. Math. Soc., Providence, RI, 1999. MR 1677884 (2000g:47048)
  • 14. P. Deift, A. R. Its, and X. 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 (98k:47097)
  • 15. 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 (2001g:42050)
  • 16. -, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), no. 12, 1491-1552. MR 1711036 (2001f:42037)
  • 17. H. Farkas and I. Kra, Riemann surfaces, Grad. Texts in Math., vol. 71, Springer-Verlag, Berlin, 1980. MR 583745 (82c:30067)
  • 18. R. Fernandez, J. Frohlich, and A. Sokal, Random walks, critical phenomena, and triviality in quantum field theory, Texts and Monogr. in Phys., Springer-Verlag, Berlin, 1992. MR 1219313 (94j:81140)
  • 19. A. S. Fokas, A. R. Its, and A. V. Kitaev, Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), no. 2, 313-344. MR 1137067 (93a:58080)
  • 20. -, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), no. 2, 395-430. MR 1174420 (93h:81115)
  • 21. P. J. Forrester, Log-gases and random matrices, London Math. Soc. Monogr. Ser., 34, Princeton Univ. Press, Princeton, NJ, 2010. MR 2641363 (2011d:82001)
  • 22. 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 (2001f:31003)
  • 23. K. T.-R. McLaughlin and P. D. Miller, The $ \bar \partial $ steepest descent method for orthogonal polynomials on the real line with varying weights, Int. Math. Res. Not. IMPN 2008, Article ID rnn075, 66 p. MR 2439564 (2010a:30010)
  • 24. L. Pastur, Spectral and probabilistic aspects of matrix models, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 207-242, Math. Phys. Stud., 19, Kluwer, Acad. Publ., Dodrecht, 1996. MR 1385683 (97b:82060)
  • 25. -, From random matrices to quasi-periodic Jacobi matrices via orthogonal polynomials, J. Approx. Theory 139 (2006), no. 1-2, 269-292. MR 2220042 (2006m:47057)
  • 26. L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Grundlehren Math. Wiss., 297, Springer-Verlag, Berlin, 1992. MR 1223779 (94h:47068)
  • 27. L. Pastur and M. Shcherbina, Eigenvalue distribution of large random matrices, Math. Surveys Monogr., 171, Amer. Math. Soc., Providence, RI, 2011. MR 2808038 (2012d:60015)
  • 28. F. Peherstorfer and P. Yuditskii, Asymptotic behavior of polynomials orthonormal on a homogeneous set, J. Anal. Math. 89 (2003), 113-154. MR 1981915 (2004j:42022)
  • 29. E. B. Saff and V. Totik, Logarithmic potentials with external fields, Grundlehren Math. Wiss., 316, Springer-Verlag, Berlin, 1997. MR 1485778 (99h:31001)
  • 30. B. Simon, Szego's theorem and its descendants. Spectral theory for $ L^{2}$ perturbations of orthogonal polynomials, Princeton Univ. Press, Princeton, NJ, 2011. MR 2743058 (2012b:47080)
  • 31. M. Sodin and P. 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 (2000k:47033)
  • 32. H. Stahl and V. Totik, General orthogonal polynomials, Encyclopedia Math. Appl., 43, Cambridge Univ. Press, Cambridge, 1992. MR 1163828 (93d:42029)
  • 33. S. Suetin, On trace formulas for a class of Jacobi operators, Mat. Sb. 198 (2007), no. 6, 107-138; English transl., Sb. Math. 198 (2007), no. 5-6, 857-885. MR 2355367 (2009e:47052)
  • 34. G. Szegő, Orthogonal polynomials, Amer. Math. Soc., Providence, RI, 1975. MR 0372517 (51:8724)
  • 35. G. Teschl, Jacobi operators and completely integrable nonlinear lattices, Math. Surveys Monogr., 72, Amer. Math. Soc., Providence, RI, 2000. MR 1711536 (2001b:39019)
  • 36. V. Totik, Weighted Approximation with Varying Weight, Lecture Notes in Math., vol. 1569, Springer-Verlag, Berlin, 1994. MR 1290789 (96f:41002)
  • 37. H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Adv. Math. 3 (1969), 127-232. MR 0239059 (39:418)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 37K40, 35Q53, 37K45, 35Q15

Retrieve articles in all journals with MSC (2010): 37K40, 35Q53, 37K45, 35Q15


Additional Information

I. Egorova
Affiliation: B. Verkin Institute for Low Temperature Physics, Lenin Avenue 47, Kharkiv 61103, Ukraine
Email: iraegorova@gmail.com

L. Pastur
Affiliation: B. Verkin Institute for Low Temperature Physics, Lenin Avenue 47, Kharkiv 61103, Ukraine
Email: pastur2001@yahoo.com

DOI: https://doi.org/10.1090/S1061-0022-2014-01287-3
Keywords: Riemann--Hilbert problem, asymptotics, varying weights
Received by editor(s): November 15, 2012
Published electronically: March 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society