Figure 1

Spectral density of a degenerate one-factor model. The gray areas are the average over 1000 numerical simulations of a one-factor model of $N=100$ variables for $T=500$ time steps. The value of $\gamma$ is 0.25. The dashed line is the theoretical prediction and the vertical lines indicate the spectrum of the model.Reuse & Permissions

Figure 2

The low part of the spectrum of the correlation matrix of a nondegenerate one-factor model (circle). In this case, $N=2000$ and the ${\gamma }_i$ are extracted from an exponential distribution with $\overline{\gamma }=0.25$. The continuous line is the prediction based on the ansatz discussed in the text.Reuse & Permissions

Figure 3

Low part of the spectral density of a one-factor model in which ${\gamma }_i$ is distributed uniformly between 0 and 1. We performed 1000 numerical simulations of a one-factor model of $N=100$ variables for $T=500$ time steps. The dashed line is the theoretical result obtained through the theory developed in the text. The dotted line is the spectral density predicted by assuming that the components of the correlation matrix—which are orthogonal to the eigenspace associated with the largest eigenvalue—are described by statistical fluctuations and are described by Eq. (9). The inset shows the model spectrum (continuous lines) and the noise dressed (dashed line) spectral densities. The vertical line indicates the largest eigenvalue of the spectrum of the model.Reuse & Permissions

Figure 4

Spectral density of a two-level hierarchical factor model. The parameters are $n_1=100$, $n_2=70$, $\Gamma =0.5$, ${\gamma }_1=0.2$, ${\gamma }_2=0.3$, and $T=500$. The gray area is based on the average over 1000 simulations and the dashed line is the theoretical result. The vertical lines indicate the position of the eigenvalues of the spectrum of the model.Reuse & Permissions

Figure 5

Spectral density of the one-factor model in with a sinusoidal factor described by Eq. (22). The parameters are $N=100$, $T=500$, $\gamma =0.5$, $\omega =1$, $\tau =1$, and the phase $\varphi$ is distributed uniformly between 0 and $2\pi$. The dashed line is the analytical result based on the RMT.Reuse & Permissions