Gernot Akemann

Higher genus correlators for the hermitian matrix model with multiple cuts

Abstract An iterative scheme is set up for solving the loop equation of the hermitian one-matrix model with a multi-cut structure. Explicit results are presented for genus one for an arbitrary but finite number of cuts. Due to the complicated form of the boundary conditions, the loop correlators now contain elliptic integrals. This demonstrates the existence of new universality classes for the hermitian matrix model. The two-cut solution is investigated in more detail, including the double scaling limit. It is shown that in special cases it differs from the known continuum solution with one cut.

Universality of random matrices in the microscopic limit and the Dirac operator spectrum

Abstract We prove the universality of correlation functions of chiral unitary and unitary ensembles of random matrices in the microscopic limit. The essence of the proof consists in reducing the three-term recursion relation for the relevant orthogonal polynomials into a Bessel equation governing the local asymptotics around the origin. The possible physical interpretation as the universality of the soft spectrum of the Dirac operator is briefly discussed.

Products of Rectangular Random Matrices: Singular Values and Progressive Scattering

We discuss the product of M rectangular random matrices with independent Gaussian entries, which have several applications, including wireless telecommunication and econophysics. For complex matrices an explicit expression for the joint probability density function is obtained using the Harish-Chandra-Itzykson-Zuber integration formula. Explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two-matrix model and the method of biorthogonal polynomials. This generalizes the classical result for the so-called Wishart-Laguerre Gaussian unitary ensemble (or chiral unitary ensemble) at M=1, and previous results for the product of square matrices. The correlation functions are given by a determinantal point process, where the kernel can be expressed in terms of Meijer G-functions. We compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices. The location of the end points of support for the latter are analyzed in detail for general M. Finally, we consider the so-called ergodic mutual information, which gives an upper bound for the spectral efficiency of a MIMO communication channel with multifold scattering.

Singular value correlation functions for products of Wishart random matrices

We consider the product of M quadratic random matrices with complex elements and no further symmetry, where all matrix elements of each factor have a Gaussian distribution. This generalizes the classical Wishart-Laguerre Gaussian unitary ensemble with M = 1. In this paper, we first compute the joint probability distribution for the singular values of the product matrix when the matrix size N and the number M are fixed but arbitrary. This leads to a determinantal point process which can be realized in two different ways. First, it can be written as a one-matrix singular value model with a nonstandard Jacobian, or second, for M >= 2, as a two-matrix singular value model with a set of auxiliary singular values and a weight proportional to the Meijer G-function. For both formulations, we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct. They are given in terms of the hypergeometric and Meijer G-functions, generalizing the Laguerre polynomials for M = 1. Our investigation was motivated from applications in telecommunication of multi-layered scattering multiple-input and multiple-output channels. We present the ergodic mutual information for finite-N for such a channel model with M - 1 layers of scatterers as an example.

Universal microscopic correlation functions for products of independent Ginibre matrices

We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n>1 and generalise the known Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.

Recent exact and asymptotic results for products of independent random matrices

In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from ensembles of independent real, complex, or quaternionic Ginibre matrices, or truncated unitary matrices. Additional mixing within one ensemble between matrices and their inverses is also covered. Exact determinantal and Pfaffian expressions are given in terms of the respective kernels of orthogonal polynomials or functions. Here we list all known cases and some straightforward generalisations. The asymptotic results for large matrix size include new microscopic universality classes at the origin and a generalisation of weak non-unitarity close to the unit circle. So far in all other parts of the spectrum the known standard universality classes have been identified. In the limit of infinite products the Lyapunov and stability exponents share the same normal distribution. To leading order they both follow a permanental point processes. Our focus is on presenting recent developments in this rapidly evolving area of research.

Equivalence of QCD in the epsilon-regime and chiral Random Matrix Theory with or without chemical potential

We prove that QCD in the epsilon-regime of chiral Perturbation Theory is equivalent to chiral Random Matrix Theory for zero and both non-zero real and imaginary chemical potential mu. To this aim we prove a theorem that relates integrals over fermionic and bosonic variables to super-Hermitian or super-Unitary groups also called superbosonization. Our findings extend previous results for the equivalence of the partition functions, spectral densities and the quenched two-point densities. We can show that all k-point density correlation functions agree in both theories for an arbitrary number of quark flavors, for either mu=0 or mu=/=0 taking real or imaginary values. This implies the equivalence for all individual k-th eigenvalue distributions which are particularly useful to determine low energy constants from Lattice QCD with chiral fermions.

Characteristic polynomials of complex random matrix models

Abstract We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their Hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for Hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite-N, where N is the size of the matrix. We give some explicit examples at finite-N for specific weight functions. The characteristic polynomials in the large-N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large-N limit.

Multicritical Microscopic Spectral Correlators of Hermitian and Complex Matrices

We find the microscopic spectral densities and the spectral correlators associated with multicritical behavior for both hermitian and complex matrix ensembles, and show their universality. We conjecture that microscopic spectral densities of Dirac operators in certain theories without spontaneous chiral symmetry breaking may belong to these new universality classes.

MATRIX MODELS AND QCD WITH CHEMICAL POTENTIAL

The random matrix model approach to quantum chromodynamics (QCD) with nonvanishing chemical potential is reviewed. The general concept using global symmetries is introduced, as well as its relation to field theory, the so-called epsilon regime of chiral perturbation theory (∊χPT). Two types of matrix model results are distinguished: phenomenological applications leading to phase diagrams, and an exact limit of the QCD Dirac operator spectrum matching with ∊χPT. All known analytic results for the spectrum of complex and symplectic matrix models with chemical potential are summarised for the symmetry classes of ordinary and adjoint QCD, respectively. These include correlation functions of Dirac operator eigenvalues in the complex plane for real chemical potential, and in the real plane for imaginary isospin chemical potential. Comparisons of these predictions to recent lattice simulations are also discussed.