Results 1  10
of
13
The largest eigenvalue of real symmetric, Hermitian and hermitian selfdual random matrix models with rank one external source, part I.” arXiv:1012.4144
, 2010
"... We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β = 1), Hermitian (β = 2), and Hermitian selfdual (β = 4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
(Show Context)
We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β = 1), Hermitian (β = 2), and Hermitian selfdual (β = 4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “oneband” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β> 0. 1 Introduction and statement of results 1.1
Random matrices with equispaced external source
, 2013
"... 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 o ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
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 the number of orthogonality weights of the polynomials grows with the degree. Nevertheless we are able to characterize them in terms of a pair of 2 × 1 vectorvalued RiemannHilbert problems, and to perform an asymptotic analysis of the RiemannHilbert problems. 1
On Finite Rank Deformations of Wigner Matrices II: Delocalized Perturbations
, 2012
"... ..."
(Show Context)
Scaling Limits of Correlations of Characteristic Polynomials for the Gaussian βEnsemble with
 External Source, Int. Math. Res. Notices
"... ar ..."
Limits of spiked random matrices II
"... The top eigenvalues of rank r spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for nearcritical perturbations, fully resolving the conjecture of Baik, Be ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The top eigenvalues of rank r spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for nearcritical perturbations, fully resolving the conjecture of Baik, Ben Arous and Péche ́ (2005). The starting point is a new (2r + 1)diagonal form that is algebraically natural to the problem; for both models it converges to a certain random Schrödinger operator on the halfline with r × r matrixvalued potential. The perturbation determines the boundary condition, and the lowlying eigenvalues describe the limit jointly over all perturbations in a fixed subspace. We treat the real, complex and quaternion (β = 1, 2, 4) cases simultaneously. We also characterize the limit laws in terms of a diffusion related to Dyson’s Brownian motion, and further in terms of a linear parabolic PDE; here β is simply a parameter. At β = 2 the PDE appears to reconcile with known Painleve ́ formulas for these rparameter deformations of the GUE TracyWidom law.
Brownian Motions with OneSided Collisions: The Stationary Case
, 2015
"... We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according t ..."
Abstract
 Add to MetaCart
(Show Context)
We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process spacetime stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finitedimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the limit t→∞. This leads to a new universal crossover process.