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}
}
  • T. Claeys, Dong Wang
  • 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… 
View on Springer
arxiv.org
24 Citations
Highly Influential Citations
4
Background Citations
9
Methods Citations
12

24 Citations

Universality for Random Matrices with Equi-spaced External Source: A Case Study of a Biorthogonal Ensemble
We prove the edge and bulk universality of random Hermitian matrices with equi-spaced external source. One feature of our method is that we use neither a Christoffel-Darboux type formula, nor a…
Edge universality for deformed Wigner matrices
We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume…
Bulk universality for deformed Wigner matrices
We consider N×N random matrices of the form H=W+V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are…
Biorthogonal ensembles with two-particle interactions
We investigate determinantal point processes on [0, +∞) of the form We prove that the biorthogonal polynomials associated with such models satisfy a recurrence relation and a Christoffel–Darboux…
Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices
We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data…
Universality of local eigenvalue statistics in random matrices with external source
Consider a random matrix of the form $W_n = M_n + D_n$, where $M_n$ is a Wigner matrix and $D_n$ is a real deterministic diagonal matrix ($D_n$ is commonly referred to as an external source in the…
A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles
Muttalib--Borodin ensembles in random matrix theory --- realisations and correlation functions
Muttalib--Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the form $\prod_{1 \le j 0$, realisations in terms of the eigenvalue…
1 2 M ay 2 01 7 MUTTALIB – BORODIN ENSEMBLES IN RANDOM MATRIX THEORY — REALISATIONS AND CORRELATION FUNCTIONS
Muttalib–Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the form ∏ 1≤j<k≤N (λk − λj)(λ θ k − λj ). We study the Laguerre and…
Asymptotic Expansion of a Partition Function Related to the Sinh-model
This paper develops a method to carry out the large-$N$ asymptotic analysis of a class of $N$-dimensional integrals arising in the context of the so-called quantum separation of variables method. We…
...
...

References

SHOWING 1-10 OF 51 REFERENCES
Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials
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…
Random Hermitian matrices in an external field
Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Supercritical and Subcritical Regimes
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
TLDR
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
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
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
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
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
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…
...
...