Random matrix ensembles for PTPT-symmetric systems

Recently much effort has been made towards the introduction of non-Hermitian random matrix models respecting $PT$-symmetry. Here we show that there is a one-to-one correspondence between complex $PT$-symmetric matrices and split-complex and split-quaternionic versions of Hermitian matrices. We introduce two new random matrix ensembles of (a) Gaussian split-complex Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary sizes. They are related to the split signature versions of the complex and the quaternionic numbers, respectively. We conjecture that these ensembles represent universality classes for $PT$-symmetric matrices. For the case of $2\times2$ matrices we derive analytic expressions for the joint probability distributions of the eigenvalues, the one-level densities and the level spacings in the case of real eigenvalues.Comment: 9 pages, 3 figures, typos corrected, small changes, accepted for publication in Journal of Physics

Sampling expansions associated with quaternion difference equations

Starting with a quaternion difference equation with boundary conditions, a parameterized sequence which is complete in finite dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of discrete transform, we form a quaternion function space whose elements have sampling expansions. Moreover, through formulating boundary-value problems, we make a connection between a class of tridiagonal quaternion matrices and polynomials with quaternion coefficients. We show that for a tridiagonal symmetric quaternion matrix, one can always associate a quaternion characteristic polynomial whose roots are eigenvalues of the matrix. Several examples are given to illustrate the results

Quaternionic R transform and non-hermitian random matrices

Using the Cayley-Dickson construction we rephrase and review the non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I. Zahed, Nucl.Phys. B $\textbf{501}$, 603 (1997)], that generalizes the free probability calculus to asymptotically large non-hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (non-crossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of $X$ and its hermitian conjugate $X^\dagger$: \langle\langle \frac{1}{N} \mbox{Tr} X^{a} X^{\dagger b} X^c \ldots \rangle\rangle for $N\rightarrow \infty$. We show that the R transform for gaussian elliptic laws is given by a simple linear quaternionic map $\mathcal{R}(z+wj) = x + \sigma^2 \left(\mu e^{2i\phi} z + w j\right)$ where $(z,w)$ is the Cayley-Dickson pair of complex numbers forming a quaternion $q=(z,w)\equiv z+ wj$. This map has five real parameters $\Re e x$, $\Im m x$, $\phi$, $\sigma$ and $\mu$. We use the R transform to calculate the limiting eigenvalue densities of several products of gaussian random matrices.Comment: 27 pages, 16 figure

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 \bY= \bA^{-1} \bB, where \bA and \bB are independent $N\times N$ 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 jpdf and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices $\beta=1,2,4$. 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 \bY= \bA^{-1} \bB, where \bA and \bB are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.Comment: 25 pages, 3 figures. Added another citation in version