Individual Eigenvalue Distributions for the Wilson Dirac Operator

We derive the distributions of individual eigenvalues for the Hermitian Wilson Dirac Operator D5 as well as for real eigenvalues of the Wilson Dirac Operator DW. The framework we provide is valid in the epsilon regime of chiral perturbation theory for any number of flavours Nf and for non-zero low energy constants W6, W7, W8. It is given as a perturbative expansion in terms of the k-point spectral density correlation functions and integrals thereof, which in some cases reduces to a Fredholm Pfaffian. For the real eigenvalues of DW at fixed chirality nu this expansion truncates after at most nu terms for small lattice spacing "a". Explicit examples for the distribution of the first and second eigenvalue are given in the microscopic domain as a truncated expansion of the Fredholm Pfaffian for quenched D5, where all k-point densities are explicitly known from random matrix theory. For the real eigenvalues of quenched DW at small "a" we illustrate our method by the finite expansion of the corresponding Fredholm determinant of size nu.Comment: 20 pages, 5 figures; v2: typos corrected, refs added and discussion of W6 and W7 extende

Weak commutation relations and eigenvalue statistics for products of rectangular random matrices

Ipsen J, Kieburg M. Weak commutation relations and eigenvalue statistics for products of rectangular random matrices. Physical Review E. 2014;89(3): 32106.We study the joint probability density of the eigenvalues of a product of rectangular real, complex, or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary, or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously has been discussed for infinite matrix size. Moreover, we derive the joint probability densities of the eigenvalues. To illustrate our results, we apply them to a product of random matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed version thereof. For these weights, we show that the product of complex random matrices yields a determinantal point process, while the real and quaternion matrix ensembles correspond to Pfaffian point processes. Our results are visualized by numerical simulations. Furthermore, we present an application to a transport on a closed, disordered chain coupled to a particle bath

Products of Rectangular Random Matrices: Singular Values and Progressive Scattering

Akemann G, Ipsen J, Kieburg M. Products of Rectangular Random Matrices: Singular Values and Progressive Scattering. Physical Review E. 2013;88(5): 52118.We discuss the product of $M$ rectangular random matrices with independentGaussian entries, which have several applications including wirelesstelecommunication and econophysics. For complex matrices an explicit expressionfor the joint probability density function is obtained using theHarish-Chandra--Itzykson--Zuber integration formula. Explicit expressions forall correlation functions and moments for finite matrix sizes are obtainedusing a two-matrix model and the method of bi-orthogonal polynomials. Thisgeneralises the classical result for the so-called Wishart--Laguerre Gaussianunitary ensemble (or chiral unitary ensemble) at M=1, and previous results forthe product of square matrices. The correlation functions are given by adeterminantal point process, where the kernel can be expressed in terms ofMeijer $G$-functions. We compare the results with numerical simulations andknown results for the macroscopic density in the limit of large matrices. Thelocation of the endpoints of support for the latter are analysed in detail forgeneral $M$. Finally, we consider the so-called ergodic mutual information,which gives an upper bound for the spectral efficiency of a MIMO communicationchannel with multi-fold scattering

Universal microscopic correlation functions for products of truncated unitary matrices

Akemann G, Burda Z, Kieburg M, Nagao T. Universal microscopic correlation functions for products of truncated unitary matrices. Journal of Physics: A Mathematical and Theoretical. 2014;47(25): 255202.We investigate the spectral properties of the product of $M$ complexnon-Hermitian random matrices that are obtained by removing $L$ rows andcolumns of larger unitary random matrices uniformly distributed on the group${\rm U}(N+L)$. Such matrices are called truncated unitary matrices or randomcontractions. We first derive the joint probability distribution for theeigenvalues of the product matrix for fixed $N,\ L$, and $M$, given by astandard determinantal point process in the complex plane. The weight howeveris non-standard and can be expressed in terms of the Meijer G-function. Theexplicit knowledge of all eigenvalue correlation functions and thecorresponding kernel allows us to take various large $N$ (and $L$) limits atfixed $M$. At strong non-unitarity, with $L/N$ finite, the eigenvalues condenseon a domain inside the unit circle. At the edge and in the bulk we find thesame universal microscopic kernel as for a single complex non-Hermitian matrixfrom the Ginibre ensemble. At the origin we find the same new universalityclasses labelled by $M$ as for the product of $M$ matrices from the Ginibreensemble. Keeping a fixed size of truncation, $L$, when $N$ goes to infinityleads to weak non-unitarity, with most eigenvalues on the unit circle as forunitary matrices. Here we find a new microscopic edge kernel that generalizesthe known results for M=1. We briefly comment on the case when each productmatrix results from a truncation of different size $L_j$