Geodesic
Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source
Teoretičeskaâ i matematičeskaâ fizika, Tome 159 (2009) no. 1, pp. 34-57 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Mots-clés : random matrix, multiple orthogonal polynomial.
Keywords: matrix model, eigenvalue distribution, Brownian bridge 
@article{TMF_2009_159_1_a1,
     author = {A. I. Aptekarev and V. G. Lysov and D. N. Tulyakov},
     title = {Global eigenvalue distribution regime of random matrices with an~anharmonic potential and an~external source},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {34--57},
     year = {2009},
     volume = {159},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2009_159_1_a1/}
}
TY  - JOUR
AU  - A. I. Aptekarev
AU  - V. G. Lysov
AU  - D. N. Tulyakov
TI  - Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2009
SP  - 34
EP  - 57
VL  - 159
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2009_159_1_a1/
LA  - ru
ID  - TMF_2009_159_1_a1
ER  - 
%0 Journal Article
%A A. I. Aptekarev
%A V. G. Lysov
%A D. N. Tulyakov
%T Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2009
%P 34-57
%V 159
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2009_159_1_a1/
%G ru
%F TMF_2009_159_1_a1
A. I. Aptekarev; V. G. Lysov; D. N. Tulyakov. Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source. Teoretičeskaâ i matematičeskaâ fizika, Tome 159 (2009) no. 1, pp. 34-57. http://geodesic.mathdoc.fr/item/TMF_2009_159_1_a1/

[1] F. Daison, Statisticheskaya teoriya energeticheskikh urovnei slozhnykh sistem, IL, M., 1963 | MR | MR | MR | Zbl

[2] M. L. Mehta, Random Matrices, Academic Press, Boston, 1991 | MR | Zbl

[3] P. J. Forrester, N. C. Snaith, J. J. M. Verbaarschot, J. Phys. A, 36:12 (2003), R1–R10 | DOI | MR | Zbl

[4] T. Nagao, P. J. Forrester, Nucl. Phys. B, 620:3 (2002), 551–565 | DOI | MR | Zbl

[5] C. Itzykson, J.-B. Zuber, J. Math. Phys., 21:3 (1980), 411–421 | DOI | MR | Zbl

[6] S. Karlin, J. McGregor, Pacific J. Math., 9:4 (1959), 1141–1164 | DOI | MR | Zbl

[7] L. A. Pastur, TMF, 10:1 (1972), 102–112 | DOI | MR

[8] E. Brézin, S. Hikami, Phys. Rev. E, 55:4 (1997), 4067–4083 | DOI | MR

[9] E. Brézin, S. Hikami, Nucl. Phys. B, 479:3 (1996), 697–706 | DOI | MR | Zbl

[10] E. Brézin, S. Hikami, Phys. Rev. E, 56:1 (1997), 264–269 | DOI

[11] E. Brézin, S. Hikami, Phys. Rev. E, 57:4 (1998), 4140–4149 | DOI | MR

[12] E. Brézin, S. Hikami, Phys. Rev. E, 58:6 (1998), 7176–7185 | DOI | MR

[13] P. M. Bleher, A. B. J. Kuijlaars, Int. Math. Res. Notices, 2004:3 (2004), 109–129 | DOI | MR | Zbl

[14] P. M. Bleher, A. B. J. Kuijlaars, Comm. Math. Phys., 252:1–3 (2004), 43–76 | DOI | MR | Zbl

[15] A. I. Aptekarev, P. M. Bleher, A. B. J. Kuijlaars, Comm. Math. Phys., 259:2 (2005), 367–389 | DOI | MR | Zbl

[16] P. M. Bleher, A. B. J. Kuijlaars, Comm. Math. Phys., 270:2 (2007), 481–517 | DOI | MR | Zbl

[17] K. T.-R. McLaughlin, Nonlinearity, 20:7 (2007), 1547–1571 | DOI | MR | Zbl

[18] A. I. Aptekarev, J. Comput. Appl. Math., 99:1–2 (1998), 423–447 | DOI | MR | Zbl

[19] A. A. Gonchar, E. A. Rakhmanov, “O skhodimosti sovmestnykh approksimatsii Pade dlya sistem funktsii markovskogo tipa”, Teoriya chisel, matematicheskii analiz i ikh prilozheniya, Tr. MIAN, 157, 1981, 31–48 | MR | Zbl

[20] A. I. Aptekarev, V. A. Kalyagin, Asimptotika kornya $n$-i stepeni iz polinomov sovmestnoi ortogonalnosti i algebraicheskie funktsii, Preprint No 60, In-t prikl. matem. im. M. V. Keldysha AN SSSR, 1986 | MR

[21] J. Nuttall, J. Approx. Theory, 42:4 (1984), 299–386 | DOI | MR | Zbl

[22] A. A. Gonchar, E. A. Rakhmanov, UMN, 40:4(244) (1985), 155–156 | MR | Zbl

[23] A. I. Aptekarev, Dokl. RAN, 442:4 (2008), 443–445 | DOI

[24] A. I. Aptekarev, A. B. J. Kuijlaars, W. Van Assche, Internat. Math. Res. Papers, 2008:4 (2008), rmp007 | MR | Zbl

[25] P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, X. Zhou, Comm. Pure Appl. Math., 52:11 (1999), 1335–1425 | 3.0.CO;2-1 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[26] A. S. Fokas, A. R. Its, A. V. Kitaev, Comm. Math. Phys., 147:2 (1992), 395–430 | DOI | MR | Zbl

[27] W. Van Assche, J. S. Geronimo, A. B. J. Kuijlaars, “Riemann–Hilbert problems for multiple orthogonal polynomials”, Special Functions, 2000: Current Perspective and Future Directions, NATO Sci. Ser. II. Math. Phys. Chem., 30, eds. J. Bustoz et al., Kluwer, Dordrecht, 2001, 23–59 | MR | Zbl