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

Tau-function theory of chaotic quantum transport with β = 1, 2, 4

We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and applies to all the symmetry classes β∈{1,2,4} of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for β = 1, 4, thus proving a number of conjectures of Khoruzhenko et al. (in Phys Rev B 80:(12)125301, 2009). We derive differential equations that characterize the cumulant generating functions for all β∈{1,2,4}. Furthermore, when β = 2 we show that the cumulant generating function of the Wigner delay time can be expressed in terms of the Painlevé III′ transcendant. This allows us to study properties of the cumulants of the Wigner delay time in the asymptotic limit n→∞. Finally, for all the symmetry classes and for any number of open channels, we derive a set of recurrence relations that are very efficient for computing cumulants at all orders