A bordering of Gaussian unitary ensemble matrices is considered, in which the bordered row consists of zero mean complex Gaussians N[0, σ/2] + iN[0, σ/2] off the diagonal and the real Gaussian N$[\mu ,\sigma /\sqrt{2}]$ on the diagonal. We compute the explicit form of the eigenvalue probability function for such matrices as well as that for matrices obtained by repeating the bordering. The correlations are in general determinantal, and in the single bordering case the explicit form of the correlation kernel is computed. In the large N limit it is shown that μ and/or σ can be tuned to induce a separation of the largest eigenvalue. This effect is shown to be controlled by a single parameter, universal correlation kernel.
REFERENCES
1.
Akhanjee
, S.
and Rudnick
, J.
, “Spherical spin-glass – Coulomb gas duality: Exact solution beyond mean-field theory
,” arXiv:1003.2715, 2010
.2.
Baik
, J.
, Ben Arous
, G.
, and Péché
, S.
, “Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
,” Ann. Prob.
33
, 1643
(2005
).3.
Bassler
, K. E.
, Forrester
, P. J.
, and Frankel
, N. E.
, “Eigenvalue separation in some random matrix models
,” J. Math. Phys.
50
, 033302
(2009
).4.
5.
Borodin
, A.
, “Biorthogonal ensembles
,” Nucl. Phys. B
536
, 704
(1998
).6.
Desrosiers
, P.
and Forrester
, P. J.
, “Hermite and Laguerre β-ensembles: Asymptotic corrections to the eigenvalue density
,” Nucl. Phys. B
743
, 307
(2006
).7.
Dyson
, F. J.
, “A Brownian motion model for the eigenvalues of a random matrix
,” J. Math. Phys.
3
, 1191
(1962
).8.
Dumitriu
, I.
and Edelman
, A.
, “Matrix models for beta ensembles
,” J. Math. Phys.
43
, 5830
(2002
).9.
Forrester
, P. J.
, “The spectrum edge of random matrix ensembles
,” Nucl. Phys. B
402
, 709
(1993
).10.
Forrester
, P. J.
, Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles, nlin.SI/0005064, 2000
.11.
Forrester
, P. J.
, Log-gases And Random Matrices
(Princeton University Press
, Princeton, NJ
, 2010
).12.
Forrester
, P. J.
and Rains
, E. M.
, “Interpretations of some parameter dependent generalizations of classical matrix ensembles
,” Probab. Theory Relat. Fields
131
, 1
(2005
).13.
14.
Kosterlitz
, J. M.
, Thouless
, D. J.
, and Jones
, R. C.
, “Spherical model of a spin glass
,” Phys. Rev. Lett.
36
, 1217
(1976
).15.
May
, R. M.
, “Will a large complex system be stable?
,” Nature (London)
238
, 413
(1972
).16.
Muttalib
, K. A.
, “Random matrix models with additional interactions
,” J. Phys. A
28
, L159
(1996
).17.
Péché
, S.
, “The largest eigenvalue of small rank perturbations of Hermitian random matrices
,” Probab. Theory Relat. Fields
134
, 127
(2006
).18.
Porter
, C. E.
, Statistical theories of spectra: Fluctuations
(Academic
, New York
, 1965
).19.
Smolyarenko
, I. E.
and Simons
, B. D.
, “Parametric spectral statistics in unitary random matrix ensembles: from distribution functions to intra-level correlations
,” J. Phys. A
36
, 3551
(2003
).© 2010 American Institute of Physics.
2010
American Institute of Physics
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