Exact spectral densities of complex noise-plus-structure random matrices

We use supersymmetry to calculate exact spectral densities for a class of complex random matrix models having the form $M=S+LXR$, where $X$ is a random noise part $X$, and $S,L,R$ are fixed structure parts. This is a certain version of the “external field” random matrix models. We find twofold integral formulas for arbitrary structural matrices. We investigate some special cases in detail and carry out numerical simulations. The presence or absence of a normality condition on $S$ leads to a qualitatively different behavior of the eigenvalue densities.

Figure 1

Construction of the multiplicity vector $\vec{n}=(1,1,2,1,2,2,2)$ from $\vec{u}=(5,2,4), \vec{v}=(2,5,4)$, and $\vec{w}=(1,3,5,2)$. The points depict groups of sizes determined by the corresponding multiplicities. Horizontal lines (both solid and dashed) are drawn along the boundaries of the groups of any of the vectors $\vec{u}, \vec{v}$, and $\vec{w}$, visualizing the construction of the merged vector $\vec{n}$.

Figure 2

Spectral density according to Eq. (31) as insets along two lines $L_1$ and $L_2$ in the complex plane, together with numerical simulations. The structural matrices are $S=\mathrm{diag}(-1,0,1+i), L=\mathrm{diag}(3/4,1)$, and $R=\mathrm{diag}(1,5/4,1)$ with multiplicity vectors of $\vec{u}=(2,1,3), \vec{v}=(2,4)$, and $\vec{w}=(2,1,3)$.

Figure 3

Left-hand side: complex plane of eigenvalues, from top to bottom for unperturbed $S=0$ (Ginibre), normal perturbation $S=10\left|1\right.〉\left.〈1\right|$, and non-normal perturbation $S=10\left|2\right.〉\left.〈1\right|$. Right-hand side: numerical simulations and analytical results for the spectral densities ${\rho }_G, {\rho }_N$, and ${\rho }_{\text{NN}}$ along the real axis line (dashed lines on the left-hand side). Numerical simulations are for matrices of size $N=4, \alpha =10$, and we set $\mu =N$.

Figure 4

A numerical simulation (top histogram) along with analytic spectral density plots of matrix ${(S+X)}^{-1}$ along two straight lines $L_1$ (bottom-left plot) and $L_2$ (bottom-right plot) for an external source setup of $S=(-2,2)$ with multiplicities $\vec{n}=(4,2)$.