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Mat. Sb., 2011, Volume 202, Number 2, Pages 3–56 (Mi msb7702)  

This article is cited in 37 scientific papers (total in 38 papers)

Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials

A. I. Aptekarev*, V. G. Lysov, D. N. Tulyakov

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

Abstract: Ensembles of random Hermitian matrices with a distribution measure defined by an anharmonic potential perturbed by an external source are considered. The limiting characteristics of the eigenvalue distribution of the matrices in these ensembles are related to the asymptotic behaviour of a certain system of multiple orthogonal polynomials. Strong asymptotic formulae are derived for this system. As a consequence, for matrices in this ensemble the limit mean eigenvalue density is found, and a variational principle is proposed to characterize this density.
Bibliography: 35 titles.

Keywords: random matrices, multiple orthogonal polynomials, strong asymptotics, matrix Riemann-Hilbert problem, extremal problems in the theory of logarithmic potentials.
* Author to whom correspondence should be addressed


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English version:
Sbornik: Mathematics, 2011, 202:2, 155–206

Bibliographic databases:

UDC: 517.53
MSC: 60B20, 42C05
Received: 28.01.2010 and 22.11.2010

Citation: A. I. Aptekarev, V. G. Lysov, D. N. Tulyakov, “Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials”, Mat. Sb., 202:2 (2011), 3–56; Sb. Math., 202:2 (2011), 155–206

Citation in format AMSBIB
\by A.~I.~Aptekarev, V.~G.~Lysov, D.~N.~Tulyakov
\paper Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials
\jour Mat. Sb.
\yr 2011
\vol 202
\issue 2
\pages 3--56
\jour Sb. Math.
\yr 2011
\vol 202
\issue 2
\pages 155--206

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    This publication is cited in the following articles:
    1. Martinez-Finkelshtein A. Silva G.L.F., “Spectral Curves, Variational Problems and the Hermitian Matrix Model With External Source”, Commun. Math. Phys.  crossref  isi
    2. A. Martínez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “Variation of the equilibrium energy and the $S$-property of stationary compact sets”, Sb. Math., 202:12 (2011), 1831–1852  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, “Padé–Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $S$-property of stationary compact sets”, Russian Math. Surveys, 66:6 (2011), 1015–1048  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. I. Aptekarev, A. Kuijlaars, “Hermite–Padé approximations and multiple orthogonal polynomial ensembles”, Russian Math. Surveys, 66:6 (2011), 1133–1199  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. A. V. Komlov, S. P. Suetin, “Widom's formula for the leading coefficient of a polynomial which is orthonormal with respect to a varying weight”, Russian Math. Surveys, 67:1 (2012), 183–185  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. E. A. Rakhmanov, S. P. Suetin, “Asymptotic behaviour of the Hermite–Padé polynomials of the 1st kind for a pair of functions forming a Nikishin system”, Russian Math. Surveys, 67:5 (2012), 954–956  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of Meixner polynomials and Christoffel-Darboux kernels”, Trans. Moscow Math. Soc., 73 (2012), 67–106  mathnet  crossref  mathscinet  zmath  elib
    8. A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight”, Trans. Moscow Math. Soc., 73 (2012), 139–159  mathnet  crossref  mathscinet  zmath  elib
    9. V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Method of interior variations and existence of $S$-compact sets”, Proc. Steklov Inst. Math., 279 (2012), 25–51  mathnet  crossref  mathscinet  isi  elib
    10. Starovoitov A.P., Kazimirov G.N., Astafeva A.V., “Asimptotika approksimatsii Ermita-Pade dlya dvukh eksponent”, Vesnik Vitsebskaga dzyarzhainaga universiteta, 6:72 (2012), 23–27  elib
    11. V. I. Buslaev, “Convergence of multipoint Padé approximants of piecewise analytic functions”, Sb. Math., 204:2 (2013), 190–222  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. A. Martínez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “A differential equation for Hermite–Padé polynomials”, Russian Math. Surveys, 68:1 (2013), 183–185  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, PFMT, 2013, no. 1(14), 81–87  mathnet
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    16. A. P. Starovoitov, “On asymptotic form of the Hermite–Pade approximations for a system of Mittag-Leffler functions”, Russian Math. (Iz. VUZ), 58:9 (2014), 49–56  mathnet  crossref
    17. T. Claeys, D. Wang, “Random matrices with equispaced external source”, Comm. Math. Phys., 328:3 (2014), 1023–1077  crossref  mathscinet  zmath  isi  elib
    18. A. B. J. Kuijlaars, G. L. F. Silva, “S-curves in polynomial external fields”, J. Approx. Theory, 191 (2015), 1–37  crossref  mathscinet  zmath  isi  elib
    19. A. P. Starovoitov, E. P. Kechko, “O lokalizatsii nulei approksimatsii Ermita–Pade eksponentsialnykh funktsii”, PFMT, 2015, no. 3(24), 84–89  mathnet
    20. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. V. M. Buchstaber, V. N. Dubinin, V. A. Kaliaguine, B. S. Kashin, V. N. Sorokin, S. P. Suetin, D. N. Tulyakov, B. N. Chetverushkin, E. M. Chirka, A. A. Shkalikov, “Alexander Ivanovich Aptekarev (on his 60th birthday)”, Russian Math. Surveys, 70:5 (2015), 965–973  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    22. Aptekarev A.I. Lopez Lagomasino G. Martinez-Finkelshtein A., “Strong asymptotics for the Pollaczek multiple orthogonal polynomials”, Dokl. Math., 92:3 (2015), 709–713  crossref  mathscinet  zmath  isi  scopus
    23. A. P. Starovoitov, E. P. Kechko, “Upper Bounds for the Moduli of Zeros of Hermite–Padé Approximations for a Set of Exponential Functions”, Math. Notes, 99:3 (2016), 417–425  mathnet  crossref  crossref  mathscinet  isi  elib
    24. A. V. Astafieva, A. P. Starovoitov, “Hermite-Padé approximation of exponential functions”, Sb. Math., 207:6 (2016), 769–791  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    25. A. P. Starovoitov, G. N. Kazimirov, M. V. Sidortsov, “Asimptotika approksimatsii Ermita–Pade eksponentsialnykh funktsii s kompleksnymi mnozhitelyami v pokazatelyakh eksponent”, PFMT, 2016, no. 2(27), 61–67  mathnet
    26. M. V. Sidortsov, N. A. Starovoitova, A. P. Starovoitov, “Ob asimptotike approksimatsii Ermita–Pade vtorogo roda dlya eksponentsialnykh funktsii s kompleksnymi mnozhitelyami v pokazatelyakh eksponent”, PFMT, 2017, no. 1(30), 73–77  mathnet
    27. Bertola M., Cafasso M., “The Kontsevich Matrix Integral: Convergence to the Painlevé Hierarchy and Stokes' Phenomenon”, Commun. Math. Phys., 352:2 (2017), 585–619  crossref  mathscinet  zmath  isi  scopus
    28. A. I. Aptekarev, G. López Lagomasino, A. Martínez-Finkelshtein, “On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials”, Russian Math. Surveys, 72:3 (2017), 389–449  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    29. A. P. Starovoitov, “Asymptotics of Diagonal Hermite–Padé Polynomials for the Collection of Exponential Functions”, Math. Notes, 102:2 (2017), 277–288  mathnet  crossref  crossref  mathscinet  isi  elib
    30. E. D. Belega, D. N. Tulyakov, “High-dimensional oscillators and Laguerre polynomials”, Russian Math. Surveys, 72:5 (2017), 965–967  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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