Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes

Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, which corresponds to shifting some of the energy levels of the physical system. We consider the case when the n×n external source matrix has two distinct real eigenvalues: a with multiplicity r and zero with multiplicity n−r. For a Gaussian potential, it was shown by Péché (Probab. Theory Relat. Fields 134:127–173, 2006) that when r is fixed or grows sufficiently slowly with n (a small-rank source), r eigenvalues are expected to exit the main bulk for |a| large enough. Furthermore, at the critical value of a when the outliers are at the edge of a band, the eigenvalues at the edge are described by the r-Airy kernel. We establish the universality of the r-Airy kernel for a general class of analytic potentials for r=O(n^g)r=\mathcal{O}(n^{\gamma}) for 0≤γ<1/12.     

Random Matrices with Equispaced External Source

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 vector-valued Riemann–Hilbert problems, and to perform an asymptotic analysis of the Riemann–Hilbert problems.     

The Largest Eigenvalue of Real Symmetric,Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source,Part I

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β=1), Hermitian (β=2), and Hermitian self-dual (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 “one-band” 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.     

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

We consider the random matrix ensemble with an external sourcedefined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a>1, we establish the universal behavior of local eigenvalue correlations in the limit n, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3×3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.Dedicated to Freeman Dyson on his eightieth birthdayThe first author was supported in part by NSF Grants DMS-9970625 and DMS-0354962.The second author was supported in part by projects G.0176.02 and G.0455.04 of FWO-Flanders, by K.U.Leuven research grant OT/04/24, and by INTAS Research Network NeCCA 03-51-6637.     

Near-Extreme Eigenvalues and the First Gap of Hermitian Random Matrices

We study the phenomenon of “crowding” near the largest eigenvalue \(\lambda _\mathrm{max}\) of random \(N \times N\) matrices belonging to the Gaussian Unitary Ensemble of random matrix theory. We focus on two distinct quantities: (i) the density of states (DOS) near \(\lambda _\mathrm{max}\) , \(\rho _\mathrm{DOS}(r,N)\) , which is the average density of eigenvalues located at a distance \(r\) from \(\lambda _\mathrm{max}\) and (ii) the probability density function of the gap between the first two largest eigenvalues, \(p_\mathrm{GAP}(r,N)\) . In the edge scaling limit where \(r = \mathcal{O}(N^{-1/6})\) , which is described by a double scaling limit of a system of unconventional orthogonal polynomials, we show that \(\rho _\mathrm{DOS}(r,N)\) and \(p_\mathrm{GAP}(r,N)\) are characterized by scaling functions which can be expressed in terms of the solution of a Lax pair associated to the Painlevé XXXIV equation. This provides an alternative and simpler expression for the gap distribution, which was recently studied by Witte et al. in Nonlinearity 26:1799, 2013. Our expressions allow to obtain precise asymptotic behaviors of these scaling functions both for small and large arguments.     

Distribution Functions for Random Variables for Ensembles of Positive Hermitian Matrices

Distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive Hermitian matrices. The inverse Fourier transform of the distribution is shown to be a Fredholm determinant of a certain operator that is an analogue of a Wiener-Hopf operator. The asymptotic formula shows that, up to the terms of order o(1), the distributions are Gaussian. Received: 5 November 1996 / Accepted: 8 January 1997     

Rank One Non-Hermitian Perturbations of Hermitian $$\beta $$-Ensembles of Random Matrices

We provide a tridiagonal matrix model and compute the joint eigenvalue density of a rank one non-Hermitian perturbation of a random matrix from the Gaussian or Laguerre \(\beta \)-ensemble.     

Convergence Rates of the Spectral Distributions of Large Random Quaternion Self-Dual Hermitian Matrices

In this paper, convergence rates of the spectral distributions of quaternion self-dual Hermitian matrices are investigated. We show that under conditions of finite 6th moments, the expected spectral distribution of a large quaternion self-dual Hermitian matrix converges to the semicircular law in a rate of \(O(n^{-1/2})\) and the spectral distribution itself converges to the semicircular law in rates \(O_p(n^{-2/5})\) and \(O_{a.s.}(n^{-2/5+\eta })\) . Those results include GSE as a special case.     

Large <Emphasis Type="Italic">n</Emphasis> Limit of Gaussian Random Matrices with External Source,Part II

We continue the study of the Hermitian random matrix ensemble with external source where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a=1, since the eigenvalues of M accumulate on two intervals for a>1, and on one interval for 0<a<1. The case a>1 was treated in Part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0<a<1. As in Part I we apply the Deift/Zhou steepest descent analysis to a 3×3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems.The first and third author are supported in part by INTAS Research Network NeCCA 03-51-6637 and by NATO Collaborative Linkage Grant PST.CLG.979738. The first author is supported in part by RFBR 05-01-00522 and the program “Modern problems of theoretical mathematics” RAS(DMS). The second author is supported in part by the National Science Foundation (NSF) Grant DMS-0354962. The third author is supported in part by FWO-Flanders projects G.0176.02 and G.0455.04 and by K.U.Leuven research grant OT/04/24 and by the European Science Foundation Program Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM) and the European Network in Geometry, Mathematical Physics and Applications (ENIGMA)     

The Spectrum of Heavy Tailed Random Matrices

Let X _N be an N → N random symmetric matrix with independent equidistributed entries. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of X _N , once renormalized by , converges almost surely and in expectation to the so-called semicircular distribution as N goes to infinity. In this paper we study the same question when P is in the domain of attraction of an α-stable law. We prove that if we renormalize the eigenvalues by a constant a _N of order , the corresponding spectral distribution converges in expectation towards a law which only depends on α. We characterize and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero. This work was partially supported by Miller institute for Basic Research in Science, University of California Berkeley.     

Large <Emphasis Type="Italic">n</Emphasis> Limit of Gaussian Random Matrices with External Source,Part III: Double Scaling Limit

We consider the double scaling limit in the random matrix ensemble with an external source

Non-Hermitian Tridiagonal Random Matrices and Returns to the Origin of a Random Walk

We study a class of tridiagonal matrix models, the q-roots of unity models, which includes the sign (q=2) and the clock (q=) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides, in the complex plane. Furthermore, the averaged traces of M ^k are integers that count closed random walks on the line such that each site is visited a number of times multiple of q. We obtain an explicit evaluation for them.     

Random Incidence Matrices: Moments of the Spectral Density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semicircle of small eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit) we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix.     

Third-Order Phase Transition: Random Matrices and Screened Coulomb Gas with Hard Walls

Journal of Statistical Physics - Consider the free energy of a d-dimensional gas in canonical equilibrium under pairwise repulsive interaction and global confinement, in presence of a volume...     

Erratum to: The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

We correct an error in Theorem 12 of the original article concerning the edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble.     

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2 × 2 matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.     

Products of Random Matrices and Generalised Quantum Point Scatterers

To every product of 2×2 matrices, there corresponds a one-dimensional Schrödinger equation whose potential consists of generalised point scatterers. Products of random matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in SL(2,?). We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.