# Random Matrices with Equispaced External Source

```@article{Claeys2012RandomMW,
title={Random Matrices with Equispaced External Source},
author={Tom Claeys and Dong Wang},
journal={Communications in Mathematical Physics},
year={2012},
volume={328},
pages={1023-1077}
}```
• Published 16 December 2012
• Mathematics
• Communications in Mathematical Physics
We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity. We obtain strong asymptotics for the multiple orthogonal polynomials associated to these models, and as a consequence for the average characteristic polynomials. One feature of the multiple orthogonal polynomials analyzed in this paper is that…
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## References

SHOWING 1-10 OF 51 REFERENCES
Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials
• Mathematics
• 2011
Ensembles of random Hermitian matrices with a distribution measure defined by an anharmonic potential perturbed by an external source are considered. The limiting characteristics of the eigenvalue
Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Supercritical and Subcritical Regimes
• Mathematics
• 2012
Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers and sample covariance matrices. We consider the case when the n×n external source
PDEs for the Gaussian ensemble with external source and the Pearcey distribution
The Pearcey distribution is shown to satisfy a fourth-order PDE with cubic nonlinearity, which also gives the PDE for the transition probability of the Pearcey process, a limiting process associated with nonintersecting Brownian motions on R.
Level spacing of random matrices in an external source
• Mathematics
• 1998
In an earlier work we had considered a Gaussian ensemble of random matrices in the presence of a given external matrix source. The measure is no longer unitary invariant and the usual techniques
On the largest eigenvalue of a Hermitian random matrix model with spiked external source II. Higher rank cases
• Mathematics
• 2011
This is the second part of a study of the limiting distributions of the top eigenvalues of a Hermitian matrix model with spiked external source under a general external potential. The case when the
UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY
• Mathematics
• 1999
We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with
On the largest eigenvalue of a Hermitian random matrix model with spiked external source I. Rank one case
• Mathematics
• 2010
Consider a Hermitian matrix model under an external potential with spiked external source. When the external source is of rank one, we compute the limiting distribution of the largest eigenvalue for
Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices
We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to
Topological recursion in enumerative geometry and random matrices
• Mathematics
• 2009
We review the method of symplectic invariants recently introduced to solve matrix models' loop equations in the so-called topological expansion, and further extended beyond the context of matrix