2.4. CPLX Φ’s eigenvalues along the imaginary axis


Once the 3-dimensional histograms of fig. (3) are available, one can plot their 2-dimensional cross-sections along the imaginary axis. This is precisely the content of fig. (5).

 



Figure 5: Histograms of the Φ’s eigenvalues along the imaginary axis

This plot represents a bunch of 11 functions PC1(y), which in turn are cross-sections along the imaginary axis of the 3-dimensional histograms PC1(z) of fig. (3). The different colored lines represent the cases N=2 (red line), N=3,4,5,6,7,8,10,12 and N=14 (light-blue line). Finally, we plotted in dark-blue the Girko’s law,

that we expect to hold for very large N, as already observed in fig. (3).
For what computational time is concerned, have a look in table. (1).

If we want to compare with the distributions RC1(y) of the REAL Ginibre ensemble, we have to resort to formula (25) in [Som07]. Note however that, in the last reference [Som07], the REAL Ginibre ensemble is normalized in a different way. In order to compare with our result we have to rescale the imaginary axis as y → N y, so that

       

(7)

We represent eq. (7) in the next fig. (6), for comparison with fig. (5).  




Figure 6: Theoretical distributions for the REAL Ginibre ensemble

This plot represents a bunch of 11 distributions RC1(Ny) on the (rescaled) REAL Ginibre ensemble. The different colored lines represent the cases from N=2 (red line), N=3,4,5,6,7,8,10,12 and N=14 (light-blue line).

It can be noted the asymptotical similarities in shapes between fig. (5) and fig. (6) as N become aventually large. This is apparent in the cuspid located in 0. The (positive) Y-displacement in fig. (5) is due to the running average performed on the bins located in the origin. Such an average is very small when the cuspid is very open (small N) and keep increasing in correspondence to narrower V-shape (large N).

From the formula (7.1.13) at page 298 of [AS72], that is

            

(8)

together with eq. (7) and the definition of the complemetary error function erf(z), one can easily arrive at

        

(9)

Such an inequality let a very few room to PC1(y), as one can see in the next fig. (7).

 



Figure 7: Bound in the REAL Ginibre ensemble’s distributions

The two bounds of eq. (9) are represented in dark-blue, together with PC1(y) (red line), for the case N=3.