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 $X^{†}$: $\langle \langle \frac{1}{N}\text{Tr}{X}^a{X}^{†b}{X}^c...\rangle \rangle$ for $N\to \infty$. We show that the $R$ transform for Gaussian elliptic laws is given by a simple linear quaternionic map $\mathcal{R}(z+wj)=x+{\sigma }^2\left(\mu e^{2i\varphi }z+wj\right)$ 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 $\text{Re}x,\text{Im}x,\varphi ,\sigma$, and $\mu$. We use the $R$ transform to calculate the limiting eigenvalue densities of several products of Gaussian random matrices.

Figure 1

Three possible propagators coming from the Gaussian part $d{\mu }_0\left(X\right)$ of the probability measure.

Figure 2

Example of a diagram consisting of all four kinds of third order vertices (light gray circles), contributing to cumulant of order 4: $\langle \langle \mathrm{Tr}{X}^2{X}^{†2}\rangle \rangle$. The big dashed circle emphasizes symmetry with respect to cyclic permutations under trace.

Figure 3

Schematic representation of the addition law in the case of sums of the Ginibre and GUE matrices, which results in an elliptic matrix.

Figure 4

Numerical eigenvalue distributions for matrices $(s\mathbb{1}+X)(t\mathbb{1}+X)100\times 100$ for (a) $s=0.50,t=0.75$, (c) $s=0.9,t=1.2$, and (e) $s=1.2,t=1.3$ compared with the theoretically evaluated ones in (b), (d), and (f), respectively. The edge of the distribution is shown on the plots (a), (c), and (e) (dark blue curve). Each histogram is made from ${10}^7$ eigenvalues.

Figure 5

Numerical eigenvalue distribution for matrices $(\mathbb{1}+E_1)(\mathbb{1}+E_2)100\times 100$ for (a) $\mu =1/3$, (c) $\mu =4/5$, and (e) $\mu =23/25$ compared with the theoretically evaluated ones in (b), (d), and (f), respectively. The edge of the distribution is shown on the plots (a), (c), and (e) (dark blue curve). Each histogram is made from ${10}^7$ eigenvalues.

Figure 6

Numerical eigenvalue distribution for matrices $(\mathbb{1}+H)(\mathbb{1}+X)$ of size (a) $50\times 50$, (b) $100\times 100$, and (c) $200\times 200$ compared with theoretical prediction for the edge of the distribution (dark blue points). Each histogram is made from ${10}^7$ eigenvalues. The eigenvalue domain is determined by closed regions given by analytic solutions of Eqs. (93) and (95).

Figure 7

Pictorial representation of the (a) propagator (D4) and (b), (c) the first two vertices generated by the perturbative expansion (D2). In this appendix we use the convention that external vertices are represented by open circles. In contrast, internal vertices (see below) are represented by filled circles. Internal vertices correspond to indices which are summed over.

Figure 8

Example of a planar Feynman diagram obtained from $V=4$ cubic vertices and $E=6$ propagators. It has $F=4$ closed lines, such as, e.g., the thick one surrounding the region 1. With each such closed line one can associate a face of a polyhedron constructed from the edges of the diagram. If the diagram is drawn on a sphere, the region 4 can be viewed as a compact region or the fourth face of a tetrahedron.

Figure 9

Examples of (a) a planar diagram and (b) a nonplanar one.

Figure 10

Pictorial representation of (a) $n\text{−}\mathrm{point}$ correlation function ${\widehat{m}}_n$ and (b) $n\text{−}\mathrm{point}$ connected correlation function ${\widehat{\kappa }}_n$.

Figure 11

Examples of diagrams contributing to four-point correlation function. The diagram (b) contributes to four-point connected correlation function.

Figure 12

Diagrammatic expansion of the resolvent, Eq. (D9). Sum over internal indices is implicit.

Figure 13

Example of a diagram generated by a resolvent. Its contribution is proportional to $g_3,g_4$, and $z^{-6}$.

Figure 14

One-line-irreducible diagrams that are components of the diagram presented in Fig. 13.

Figure 15

Pictorial representation of the first Dyson-Schwinger equation (D12). Each diagram may be constructed by aligning some one-line-irreducible diagrams and connecting them alternately with dashed lines.

Figure 16

Pictorial representation of the second Dyson-Schwinger equation (D13). Each one-line-irreducible diagram may be constructed by aligning some diagrams next to each other and straddling them with a connected diagram.