Universality in chaos: Lyapunov spectrum and random matrix theory

We propose the existence of a new universality in classical chaotic systems when the number of degrees of freedom is large: the statistical property of the Lyapunov spectrum is described by random matrix theory. We demonstrate it by studying the finite-time Lyapunov exponents of the matrix model of a stringy black hole and the mass-deformed models. The massless limit, which has a dual string theory interpretation, is special in that the universal behavior can be seen already at $t=0$, while in other cases it sets in at late time. The same pattern is demonstrated also in the product of random matrices.

Figure 1

The separation distribution $P\left(s\right)$ for the D0-brane matrix model (1) with $N=4,6,8,12,16$ at $t=0$ (a), and $N=4,6,8$ at $t=10$ (b). $P\left(s\right)$ agrees with $P_{\mathrm{GOE}}\left(s\right)$ at large $N$.

Figure 2

The separation distribution $P\left(s\right)$ for the D0-brane matrix model (1) with the mass deformation, $m=3, N=4,6,8,12,16$ at $t=0$ (a), and $N=4,6,8$ at $t=10$ (b). At $m\ne 0$, although $P\left(s\right)$ and $P_{\mathrm{GOE}}\left(s\right)$ do not agree at $t=0$, they become very close at $t=10$.

Figure 3

(a) Mass dependence of the difference between the mass-deformed model and the GOE random matrix, $\int ds|P\left(s\right)-P_{\mathrm{GOE}}\left(s\right)|$. The sample size is 12 000 for $N=4,6,8$ and at least 1000 (230) for $N=12$ (16), respectively. (b) Time dependence of the difference, $m=3, N=4,6,8$, with the same quantity plotted against $1/t$ in the inset (c). The difference oscillates and gradually decreases. At $N=8$, the decrease at late time is $\sim 1/t$.

Figure 4

The SFF $g\left(\tau \right)$, at $\beta =0$ for the unfolded Lyapunov spectrum of the D0-brane matrix model (1) with $N=8$ (left) and $N=24$ (right) at $t=0$ and for the unfolded eigenvalues of Gaussian random symmetric matrices with dimension $K=16(N^2-1)$.

Figure 5

The SFF $g\left(\tau \right)$ for the unfolded Lyapunov spectra of the mass-deformed model with $N=8$ and $m=3$ for $t=1$ and 10, and for the unfolded Gaussian random symmetric matrix eigenvalues with $K=16(N^2-1)=1008$.

Figure 6

(a) The difference from GOE, $\int ds|P\left(s\right)-P_{\mathrm{GOE}}\left(s\right)|$, at $t=1$, as a function of $h/\sqrt{K}$. We can see that the difference converges to an $O\left(1\right)$ value when $h/\sqrt{K}$ is fixed. (b) The same quantity for various $K$ and $t$, with $h/\sqrt{K}=1/2$. A clear convergence to GOE at large $K$ and large $t$ can be seen.

Figure 7

The average nearest-neighbor gap ratio $\langle r\rangle$ plotted against the inverse of the number of multiplied matrices, $1/t$, for the complex and real random matrix products with $K=900$ and $h=16,13,10$. The sample size is 1000 for all cases. The values for GUE and GOE random matrix eigenvalues from Ref. [35] are also shown by horizontal lines for comparison.

Figure 8

(a, b) $g\left(\tau \right)={\left|Z\left(\tau \right)\right|}^2/{\left|Z(\tau =0)\right|}^2$ for the finite-time Lyapunov exponents obtained from the singular values of $t$ real (complex) random matrix products with $K=3600$ and $h=32$, compared against $g_{\mathrm{GOE}\left(\mathrm{GUE}\right)}\left(\tau \right)=|Z_{\mathrm{GOE}\left(\mathrm{GUE}\right)}{\left(\tau \right)|}^2/{\left|Z_{\mathrm{GOE}\left(\mathrm{GUE}\right)}(\tau =0)\right|}^2$ obtained from GOE (GUE) random matrices of the same dimension $K$. We have used the unfolded spectrum. See Ref. [36] for the details of the unfolding.

Figure 9

The histogram $\rho \left(\lambda \right)$ of the local ($t=0$) Lyapunov exponents ($\lambda >0$) for the D0-brane matrix model with $m=0$ (a) and 3 (b), $N=4,6,8,12,16$. The bin width is $\mathrm{\Delta }\lambda =0.01$. The same set of data is used for the left panels of Figs. 1 and 2.

Figure 10

The histogram $\rho \left(\lambda \right)$ of the Lyapunov exponents for the D0-brane matrix model at $t=10$ for $m=0$ (a) and 3 (b), $N=4,6,8$. The bin width is $\mathrm{\Delta }\lambda =0.005$. The same set of data is used for the right panels of Figs. 1 and 2.