Correlation Functions of Ensembles of Asymmetric Real Matrices

We give a closed form for the correlation functions of ensembles 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 also derive closed forms for the matrix kernel and correlation functions for Ginibre’s real ensemble. Difficulties arise in the study of ensembles of asymmetric random matrices which do not occur in the more commonly studied ensembles of random matrices (e.g. Hermitian, unitary, complex asymmetric, etc.). These difficulties stem from the basic fact that the eigenvalues of real matrices can either be real or members of complex conjugate pairs. Due to this additional complexity, our understanding of ensembles of real asymmetric matrices lags far behind our understanding of the Hermitian ensembles. In spite of these complications many important quantities such as a closed form for ensemble averages and correlation functions can be expressed using analogous structures for their Hermitian counterparts. Here we will show that the correlation functions for an ensemble of asymmetric real matrices can be expressed as the Pfaffian of a block matrix whose entries are expressed in terms of a 2 × 2 matrix kernel associated to the ensemble. We find much inspiration from Tracy and Widom’s paper on correlation and cluster functions of Hermitian and related ensembles [19]. However, instead of using properties of the Fredholm determinant to calculate the correlation functions via the cluster functions, we use the notion of the Fredholm Pfaffian to determine the correlation functions directly. A Pfaffian analog of the Cauchy-Binet formula introduced by Rains [10] lies at the heart of our proof. For completeness we will include Rains’ proof here. In place of the identities of de Bruijn [6] used by Tracy and Widom we will use an identity of the second author [16] to compute the correlation functions for ensembles of asymmetric matrices. Rains’ Cauchy-Binet formula has applicability in a wider context than just the determination of the correlation function for ensembles of asymmetric matrices, and we will use it to give a simplified proof of the correlation function of Hermitian ensembles of random matrices when β = 1 and β = 4. Next we will apply our theory to Ginibre’s ensemble of real asymmetric matrices. Matrices in this ensemble are square (say N × N), asymmetric with iid Gaussian entries. By scaling the eigenvalues by √ N , as N → ∞, the support of the eigenvalues converges to the unit disk. This scaling allows us to analyze the asymptotics of the matrix kernel in the bulk, and we will discover a new universality class of matrix models. This will yield a closed form for the limiting correlations of points in the bulk. Two different kernels emerge: one when expanding about points on the real axis and another when expanding about point in the (non-real) complex bulk. In the latter case we find that the limiting (large N) correlation