Statistical properties of structured random matrices

Spectral properties of Hermitian Toeplitz, Hankel, and Toeplitz-plus-Hankel random matrices with independent identically distributed entries are investigated. Combining numerical and analytic arguments it is demonstrated that spectral statistics of all these low-complexity random matrices is of the intermediate type, characterized by: (i) level repulsion at short distances, (ii) an exponential decrease in the nearest-neighbor distributions at long distances, (iii) a nontrivial value of the spectral compressibility, and (iv) the existence of nontrivial fractal dimensions of eigenvectors in Fourier space. Our findings show that intermediate-type statistics is more ubiquitous and universal than was considered so far and open a new direction in random matrix theory.

Figure 1

Configuration of eigenvalues for the $P_n\left(s\right)$ correlation function.

Figure 2

Mean normalized densities $\overline{\rho }\left(\varepsilon \right)$ for different classes of structured matrices. From top to bottom at $\epsilon =0$: real and complex Toeplitz (black), independent real Toeplitz-plus-Hankel (red), independent complex Toeplitz-plus-Hankel (green), and Hankel (blue).

Figure 3

$P_n\left(s\right)$ for $0\le n\le 5$ for (top) complex Toeplitz matrices (open black squares) and (bottom) special $T+H$ matrices (filled black circles) and special $T-H$ matrices (open black squares) given by (16). Black dashed lines are $\gamma$ distributions (13) with ${\gamma }_n=2n+1$ (semi-Poisson distribution). Black solid lines are $\gamma$ distributions fitted with a single fitting parameter ${\gamma }_n$. The corresponding fitted values of ${\gamma }_n$ are given in Table 2 and plotted in the insets (filled black circles) together with the semi-Poisson prediction ${\gamma }_n=2n+1$ (solid black line).

Figure 4

(Top) $P_n\left(s\right)$ for $0\le n\le 5$ for Hankel matrices (open black circles) and real Toeplitz-plus-Hankel matrices (filled black circles). The black dashed lines are the $\gamma$ distributions (13) with ${\gamma }_n=3n+1$. The inset displays the values of ${\gamma }_n$ obtained by a one-parameter fit (same symbols as in the main panel) and given in Table 2, and the black solid line is the prediction ${\gamma }_n=3n+1$. (Bottom) The same for complex Toeplitz-plus-Hankel matrices (filled black circles). The black dashed lines are the $\gamma$ distributions (13) with ${\gamma }_n=4n+2$, and the black solid lines are the $\gamma$ distributions obtained by a one-parameter fit. The inset shows the fitted values of ${\gamma }_n$ (filled black circles) and the prediction ${\gamma }_n=4n+2$.

Figure 5

The form factor of (left) complex Toeplitz matrices (filled black circles), (right) special $T+H$ matrices (filled black circles), and special $T-H$ matrices (open black circles). Circles are the numerical computations using (57), and the solid black line is the form factor of the $\gamma$ distribution with ${\gamma }_n=2n+1$ (i.e., the semi-Poisson distribution) calculated from (49) and (50).

Figure 6

(left) The form factor of Hankel matrices (filled black circles) and independent real Toeplitz-plus-Hankel matrices (open black squares). The inset: evolution of the form factor near the origin for Hankel matrices with increasing matrix dimension $N=2^n$ with (from right to left at small $\tau$) $n=10$ (black), $n=11$ (green), $n=12$ (blue), and $n=13$ (red). (right) The form factor of independent complex Toeplitz-plus-Hankel matrices (filled black circles). The solid black line is the form factor of $\gamma$ distribution with: (left) ${\gamma }_n=3n+1$, and (right) ${\gamma }_n=4n+2$ calculated from (49) and (50). The dashed line is the form factor for short-range plasma model with the same value of ${\gamma }_n$, given by (B9) and (B10).

Figure 7

Fractal dimensions in the Fourier space for (left) complex Toeplitz matrices (solid black line) and for special $T+H$ matrices (filled black circles) and $T-H$ matrices (open black circles) given by (16) and (right) Hankel matrices (filled black circles), independent real Toeplitz-plus-Hankel matrices (open black squares), and independent complex Toeplitz-plus-Hankel matrices (solid black line). Points corresponding to $D_1$ are indicated by thin straight lines. In the insets ${\mathrm{\Delta }}_q$ and ${\mathrm{\Delta }}_{1-q}$ are plotted together for (left) complex Toeplitz matrices (solid line is ${\mathrm{\Delta }}_q$ and dashed line is ${\mathrm{\Delta }}_{1-q}$) and for (right) Hankel matrices (${\mathrm{\Delta }}_q$ is indicated by solid line and ${\mathrm{\Delta }}_{1-q}$ by dashed line), and for independent complex Toeplitz-plus-Hankel matrices (${\mathrm{\Delta }}_q$ is indicated by filled black circles connected by a thin line and ${\mathrm{\Delta }}_{1-q}$ by dashed-dot line).