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

This article is cited in 30 scientific papers (total in 31 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.

DOI: https://doi.org/10.4213/sm7702

Full text: PDF file (1195 kB)
References: PDF file   HTML file

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

Bibliographic databases:

Document Type: Article
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

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
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    2. 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
    3. 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
    4. 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
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    8. 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
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    16. T. Claeys, D. Wang, “Random matrices with equispaced external source”, Comm. Math. Phys., 328:3 (2014), 1023–1077  crossref  mathscinet  zmath  isi  elib
    17. 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
    18. A. P. Starovoitov, E. P. Kechko, “O lokalizatsii nulei approksimatsii Ermita–Pade eksponentsialnykh funktsii”, PFMT, 2015, no. 3(24), 84–89  mathnet
    19. 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
    20. 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
    21. 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
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