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

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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 
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     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},
     publisher = {mathdoc},
     volume = {159},
     number = {1},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2009_159_1_a1/}
}
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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/