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.     

Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

In the recent publication (E. Kanzieper and G. Akemann in Phys. Rev. Lett. 95:230201, 2005), an exact solution was reported for the probability p _n,k to find exactly k real eigenvalues in the spectrum of an n×n real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined.     

Singular Values of Products of Ginibre Random Matrices,Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M = 2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy–Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.     

A Real Quaternion Spherical Ensemble of Random Matrices

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form Y=A ^?1 B, where A and B are independent N×N matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue joint probability density function and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices β=1,2,4. We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure Y=A ^?1 B, where A and B are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.     

Scaling Limits of Random Skew Plane Partitions with Arbitrarily Sloped Back Walls

The paper studies scaling limits of random skew plane partitions confined to a box when the inner shapes converge uniformly to a piecewise linear function V of arbitrary slopes in [−1, 1]. It is shown that the correlation kernels in the bulk are given by the incomplete Beta kernel, as expected. As a consequence it is established that the local correlation functions in the scaling limit do not depend on the particular sequence of discrete inner shapes that converge to V. A detailed analysis of the correlation kernels at the top of the limit shape, and of the frozen boundary is given. It is shown that depending on the slope of the linear section of the back wall, the system exhibits behavior observed in either Okounkov and Reshetikhin (Commun Math Phys 269(3):571–609, 2007) or Boutillier et al. ( [math-ph], 2009).     

Scaling Limits of Waves in Convex Scalar Conservation Laws Under Random Initial Perturbations

We study waves in convex scalar conservation laws under noisy initial perturbations. It is known that the building blocks of these waves are shock and rarefaction waves, both are invariant under hyperbolic scaling. Noisy perturbations can generate complicated wave patterns, such as diffusion process of shock locations. However we show that under the hyperbolic scaling, the solutions converge in the sense of distribution to the unperturbed waves. In particular, randomly perturbed shock waves move at the unperturbed velocity in the scaling limit. Analysis makes use of the Hopf formula of the related Hamilton-Jacobi equation and regularity estimates of noisy processes. AMS subject classifications: 35L60, 35B40, 60H15     

Symmetry Properties of the k-Body Embedded Unitary Gaussian Ensemble of Random Matrices

We extend the recent study of the k-body embedded Gaussian ensembles by L. Benet, T. Rupp, and H. A. Weidenmüller (2001, Benet, Phys. Rev. Lett.87, 101601-1 and 2001, Ann. Phys. (N.Y.)292, 67) and by T. Asaga, L. Benet, T. Rupp, and H. A. Weidenmüller (cond-mat/0107363 and cond-mat/0107364). We show that central results of these papers can be derived directly from the symmetry properties of both the many-particle states and the random k-body interaction. We offer new insight into the structure of the matrix of second moments of the embedded ensemble and of the supersymmetry approach. We extend the concept of the embedded ensemble and define it purely group-theoretically.     

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.     

Determinants of Airy Operators and Applications to Random Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener–Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.     

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 Matrices: Universality of Local Eigenvalue Statistics up to the Edge

This is a continuation of our earlier paper (Tao and Vu, , 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in Tao and Vu (, 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov (Commun Math Phys 207(3):697–733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.     

A Factorization of Determinant Related to Some Random Matrices

We consider the expectation of the determinant det(–X)^–1for Im >0 associated with some random N×Nmatrices and factorize it into NStieltjes transforms of probability measures. Moreover, using this factorization, we investigate the limiting behavior of the logarithm of the quantity as N.     

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.     

Erratum to: Number Rigidity in Superhomogeneous Random Point Fields

Journal of Statistical Physics -