2.2. Distribution of the CPLX Φ’s Eigenvalues

 



Figure 2: Histograms of Φ’s Eigenvalues

This animated gif represents a sequence of 3 histograms of the complex eigenvalues of superoperators Φ, distributed according to the measure induced by running the Hans-Jürgen procedure. The sequence displays the cases N=2 (more flat), N=3 and last N=4 (more peaked).

Please note that REAL eigenvalues are excluded here… just complex eigenvalues are considered… In other words the domain is just given by the unit circle subtracted by the real segment [-1,1].

Such a domain has been divided into by 35×35 lattice, frequency count in the 1225 bins has been performed, and the probability histograms have been obtained by rescaling.

Fig. (2) displays a concentration phenomenon. The higher N, the most concentrated the cloud of eigenvalues toward the origin.

In order to keep constant the size of the cloud, we have to rescale Φ, according to

                                    (6)

Let now focus on the NC complex eigenvalues of Φ in the next fig. (3).

 



Figure 3: Histograms of Φ’s Eigenvalues

This animated gif represents a sequence of 11 histograms of the complex eigenvalues of rescaled superoperators Φ, distributed according to the measure induced by running the Hans-Jürgen procedure. The sequence displays the cases N=2 (two symmetric bumps), N=3,4,5,6,7,8,10,12,14 and finally the so-called Girko’s law,

that we expect to hold for very large N (in the plot we indicate such slide with N=infinity)

Please note that REAL eigenvalues are excluded, as in fig. (2). The domain is now enlarged to the square [-1.2,1.2]2 subtracted by the real segment [-1.2,1.2].

Such a domain has been sub-divided by a 35×35 lattice, frequency count in the 1225 bins has been performed, and the probability histograms PC1 (z) have been obtained by rescaling.