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

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Teoreticheskaya i Matematicheskaya Fizika, 2009, Volume 159, Number 1, Pages 34–57
DOI: https://doi.org/10.4213/tmf6331 (Mi tmf6331)

This article is cited in 19 scientific papers (total in 20 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

Full-text PDF (574 kB) Citations (20)

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

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.

Received: 07.07.2008

English version:
Theoretical and Mathematical Physics, 2009, Volume 159, Issue 1, Pages 448–468
DOI: https://doi.org/10.1007/s11232-009-0036-0

Bibliographic databases:

Language: Russian

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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This publication is cited in the following 20 articles:

A.V. Tsvetkova, “Real Semiclassical Approximation for the Asymptotics of Jacobi Polynomials Given by a Difference Equation”, Russ. J. Math. Phys., 31:4 (2024), 774
Martinez-Finkelshtein A., Silva G.L.F., “Spectral Curves, Variational Problems and the Hermitian Matrix Model With External Source”, Commun. Math. Phys., 383:3 (2021), 2163–2242
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
M. A. Lapik, D. N. Tulyakov, “On expanding neighborhoods of local universality of Gaussian unitary ensembles”, Proc. Steklov Inst. Math., 301 (2018), 170–179
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
A. P. Starovoitov, “Asymptotics of Diagonal Hermite–Padé Polynomials for the Collection of Exponential Functions”, Math. Notes, 102:2 (2017), 277–288
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
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.
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
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
A. P. Starovoitov, E. P. Kechko, “O lokalizatsii nulei approksimatsii Ermita–Pade eksponentsialnykh funktsii”, PFMT, 2015, no. 3(24), 84–89
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
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.
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
A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, PFMT, 2013, no. 1(14), 81–87
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
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
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
A. I. Aptekarev, A. Kuijlaars, “Hermite–Padé approximations and multiple orthogonal polynomial ensembles”, Russian Math. Surveys, 66:6 (2011), 1133–1199
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

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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