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TMF, 2009, Volume 159, Number 1, Pages 34–57 (Mi tmf6331)  

This article is cited in 18 scientific papers (total in 19 papers)

Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source

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

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

Abstract: We consider ensembles of random Hermitian matrices with a distribution measure determined by a polynomial potential perturbed by an external source. We find the genus-zero algebraic function describing the limit mean density of eigenvalues in the case of an anharmonic potential and a diagonal external source with two symmetric eigenvalues. We discuss critical regimes where the density support changes the connectivity or increases the genus of the algebraic function and consequently obtain local universal asymptotic representations for the density at interior and boundary points of its support (in the generic cases). The investigation technique is based on an analysis of the asymptotic properties of multiple orthogonal polynomials, equilibrium problems for vector potentials with interaction matrices and external fields, and the matrix Riemann–Hilbert boundary value problem.

Keywords: random matrix, matrix model, eigenvalue distribution, Brownian bridge, multiple orthogonal polynomial

DOI: https://doi.org/10.4213/tmf6331

Full text: PDF file (574 kB)
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English version:
Theoretical and Mathematical Physics, 2009, 159:1, 448–468

Bibliographic databases:

Received: 07.07.2008

Citation: A. I. Aptekarev, V. G. Lysov, D. N. Tulyakov, “Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source”, TMF, 159:1 (2009), 34–57; Theoret. and Math. Phys., 159:1 (2009), 448–468

Citation in format AMSBIB
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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:
    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. I. Aptekarev, V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants”, Sb. Math., 201:2 (2010), 183–234  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. I. Aptekarev, V. G. Lysov, D. N. Tulyakov, “Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials”, Sb. Math., 202:2 (2011), 155–206  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Bleher P., Delvaux S., Kuijlaars A.B.J., “Random matrix model with external source and a constrained vector equilibrium problem”, Comm. Pure Appl. Math., 64:1 (2011), 116–160  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    5. A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. 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
    7. A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, PFMT, 2013, no. 1(14), 81–87  mathnet
    8. 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
    9. A. P. Starovoitov, E. P. Kechko, “O lokalizatsii nulei approksimatsii Ermita–Pade eksponentsialnykh funktsii”, PFMT, 2015, no. 3(24), 84–89  mathnet
    10. 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
    11. M. A. Lapik, “Ekstremalnyi funktsional dlya vektornykh zadach ravnovesiya logarifmicheskogo potentsiala vo vneshnem pole s matritsei vzaimodeistviya Anzhelesko”, Preprinty IPM im. M. V. Keldysha, 2015, 083, 23 pp.  mathnet
    12. 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
    13. 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
    14. 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
    15. 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
    16. A. P. Starovoitov, E. P. Kechko, “On Some Properties of Hermite–Padé Approximants to an Exponential System”, Proc. Steklov Inst. Math., 298 (2017), 317–333  mathnet  crossref  crossref  isi  elib
    17. M. A. Lapik, D. N. Tulyakov, “Raspredelenie nulei mnogochlenov Ermita vblizi nulya i gaussovskie unitarnye ansambli”, Preprinty IPM im. M. V. Keldysha, 2017, 129, 11 pp.  mathnet  crossref
    18. M. A. Lapik, D. N. Tulyakov, “On expanding neighborhoods of local universality of Gaussian unitary ensembles”, Proc. Steklov Inst. Math., 301 (2018), 170–179  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    19. Martinez-Finkelshtein A., Silva G.L.F., “Critical Measures For Vector Energy: Asymptotics of Non-Diagonal Multiple Orthogonal Polynomials For a Cubic Weight”, Adv. Math., 349 (2019), 246–315  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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