Large n Limit of Gaussian Random Matrices with External Source, Part I

Bleher, Pavel; Kuijlaars, Arno B. J.

doi:10.1007/s00220-004-1196-2

Cited by 132 publications

(300 citation statements)

References 29 publications

Supporting

Mentioning

278

Contrasting

Unclassified

Order By: Relevance

“…As in [7,Section 9], we can show that E −1 (y n )R −1 (y n )R(x n )E(x n ) → I. Similar calculations give the same result if x and/or y are positive.…”

Section: Proof Of Theorem 24supporting

confidence: 82%

See 2 more Smart Citations

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

Kuijlaars

Martínez–Finkelshtein

Wielonsky

2008

Commun. Math. Phys.

120

View full text Add to dashboard Cite

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t * the smallest paths hit the hard edge and from then on are stuck to it. For t = t * we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

show abstract

“…As in [7,Section 9], we can show that E −1 (y n )R −1 (y n )R(x n )E(x n ) → I. Similar calculations give the same result if x and/or y are positive.…”

Section: Proof Of Theorem 24supporting

confidence: 82%

“…Then it may be shown (see (9.7) and [7]) that E −1 (y n )R −1 (y n )R(x n )E(x n ) → I, and we arrive at …”

Section: Proof Of Theorem 24mentioning

confidence: 73%

See 1 more Smart Citation

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

Kuijlaars

Martínez–Finkelshtein

Wielonsky

2008

Commun. Math. Phys.

120

View full text Add to dashboard Cite

show abstract

“…The case lim N →∞ k N N = α ∈ (0, 1) will be the object of a subsequent paper, and is not examined here. In this context, the limiting statistics of extreme eigenvalues are determined in [2], when W N = diag (a, . .…”

Section: A Large Rank Perturbationmentioning

confidence: 99%

The largest eigenvalue of small rank perturbations of Hermitian random matrices

Péché

2005

Probab. Theory Relat. Fields

106

View full text Add to dashboard Cite

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. We consider random Hermitian matrices with independent Gaussian entries M ij , i ≤ j with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large N limit, various limiting distributions depending on both the eigenvalues of the matrix (EM ij ) N i,j=1and its rank. This rank is also allowed to increase with N in some restricted way.

show abstract

“…the hermitian random matrix with i.i.d. (modulo symmetry) entries), in [2,3] for H (0) n being the matrix with only two eigenvalues ±a of equal multiplicity, and in [18] under the certain rather weak conditions both for random and non-random H (0) n . The local edge regime, which deals with the behavior of the eigenvalues near the edges of the spectrum (see a definition below), is also studied for many ensembles of random matrices (see e.g.…”

Section: Introductionmentioning

confidence: 99%

On Universality of Local Edge Regime for the Deformed Gaussian Unitary Ensemble

Shcherbina

2011

J Stat Phys

View full text Add to dashboard Cite

We consider the deformed Gaussian ensemble H n = H (0) n +M n in which H (0) n is a hermitian matrix (possibly random) and M n is the Gaussian unitary random matrix (GUE) independent of H (0) n . Assuming that the Normalized Counting Measure ofn converges weakly (in probability if random) to a non-random measure N (0) with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of H n .

show abstract

Large n Limit of Gaussian Random Matrices with External Source, Part I

Cited by 132 publications

References 29 publications

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

The largest eigenvalue of small rank perturbations of Hermitian random matrices

On Universality of Local Edge Regime for the Deformed Gaussian Unitary Ensemble

Contact Info

Product

Resources

About