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 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 (noncrossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of X and its Hermitian conjugate X†: 〈〈1/NTrX(a)X(†b)X(c)...〉〉 for N→∞. We show that the R transform for Gaussian elliptic laws is given by a simple linear quaternionic map R(z+wj)=x+σ(2)(μe(2iϕ)z+wj) where (z,w) is the Cayley-Dickson pair of complex numbers forming a quaternion q=(z,w)≡z+wj. This map has five real parameters Rex, Imx, ϕ, σ, and μ. We use the R transform to calculate the limiting eigenvalue densities of several products of Gaussian random matrices.