We have adopted
as a statistical tool the two-point correlation function (TPCF),
defined as the excess, due to clustering, of the probability 
of
finding a cloud in a volume 
at a distance r from another
cloud:
where 
is the average space density of the clouds as a function
of z. The TPCF is known to be a satisfactory
estimator when used to investigate weak clustering on scales considerably
smaller than the total interval covered by the data. The binning,
intrinsic to this method, causes a loss of information, but the ease in
visualizing its results and in
including observational effects in the computing codes have made of the
TPCF one of the favorite statistical estimators in cosmology.
In practice the observations provide the redshifts of the absorption lines that, due to peculiar motions, are not immediately transformed into comoving distances. Therefore the TPCF is generally computed in the velocity space, making use of the formula (Peebles 1980)
where 
is the number of observed line pairs with velocity
separations between v and 
and 
is the number
of pairs expected in the same interval from a random distribution in
redshift.
At the small velocity separations we are dealing with, the variation of the distance scale with cosmic time can be neglected and the velocity difference can be simply deduced from the redshift difference (Sargent et al. 1980)
where 
is the velocity of one cloud as measured by an observer
in the rest frame of the other.
In our line sample is obtained by averaging 5000 numerical
simulations of the observed number of redshifts, trying to account for all
the relevant cosmological and observational effects. In particular the
set of redshifts is randomly generated in the same redshift interval as
the data according to the cosmological distribution 
, where 
has been taken equal to -1.2
(Sargent
et al. 1988).
Observed pairs with a velocity splitting 
have
been merged, while simulated ones have been excluded in the estimate of
, because of the intrinsic line blending due to the typical
widths of the C IV lines.
Figure 9: Two point correlation function at large velocity separations
for the C IV systems. The dotted line shows the 95% confidence level
Figure 10: Two point correlation function at small velocity separations
for the C IV systems. The dotted line shows the 95% confidence level
The TPCF for the C IV clouds is presented in
Fig. 9 (click here) with 60 km s-1 bins. A strong clustering signal
is detected at small velocity separations (
km s-1).
At larger scales no significant signal is found, in particular the peaks
observed in the TPCF at 
km s-1,
and
at 
km s-1 are
aliases corresponding to the coupling of the low (high
frequency) power with the window function.
The present data have adequate resolution to allow a further
investigation of the distribution on scales smaller than 1000 km s-1.
The TPCF for velocity separations in the range 
is
shown in Fig. 10 (click here). In this velocity interval 48 pairs are
observed, while 
are predicted for a homogeneous distribution.
Similar results, with a significant correlation on scales up to
,
have been obtained in the works by
Petitjean & Bergeron
(1994), Womble et al. (1996) and
Songaila & Cowie (1996), carried out at
comparable resolution.
The velocity scale at which the maximum clustering signal is observed is comparable with the extension of the complex metal absorption features in the spectrum. This suggests, as already noticed by Petitjean & Bergeron (1994), that we are not seeing clustering of "galaxies'' but of gas clouds within the same galactic halo.
Other authors (Sargent et al. 1988;
Heisler et al. 1989) report, on the
basis of large samples of QSOs observed at relatively low resolution (a
few Angstrom), significant correlation of C IV lines up to scales of
. If the resolution of the present data is degraded
to the level typical of those investigations (e.g. a resolution of 
), a compatible result is
obtained, with 
.
The correlation on scales larger than 1000 km s-1 reported by Heisler et al. (1989) is not reproduced. It has to be noted, however, that it appears to be the result of the inclusion in their sample of one "exceptional'' object, 0237-233.
It has long been assessed that a correlation exists between the equivalent width of metal absorption systems and their number of components (Wolfe 1986; Petitjean & Bergeron 1990; Petitjean & Bergeron 1994).
In Fig. 11 (click here) the total equivalent width is plotted versus the number of components for the C IV absorption systems. The dashed line represents the best linear fit for all the points.
The number of components, however, is highly dependent on the spectral resolution and on subjective taste. To overcome this problem, we examined the maximum velocity separation in a given system versus the total equivalent width (Fig. 12 (click here)). A correlation is apparent and the dashed line is, again, the best linear fit.
The observed trends suggest that the clustering amplitude of the C IV lines could be a function of the column density.
In order to further investigate this issue, in
Fig. 13 (click here) the amplitude of the TPCF
(in the bin 
90 km s-1) is plotted as a
function of the median value of the C IV column density.
The upper right point represents the result obtained in this work
for the C IV absorptions in the enlarged sample.
The lower left point comes from the lower column density sample
obtained by Songaila and Cowie (1996) for the QSO 1422+231. The
TPCF has been computed, also for the latter data, according to the procedure
described in Sect. 7.1.
Lines of lower column density show indeed a smaller amplitude
of the TPCF at 
km s-1.
The correlation of the clustering amplitude
with column density is analogous to what has been observed for 
(Cristiani et al. 1997) and it is qualitatively
consistent with a picture of gravitationally induced correlations.
Figure 11: Total equivalent width of C IV systems
versus the corresponding number of components. The dashed line is the
linear correlation best fit for all points
Figure 12: Total equivalent width of C IV systems
versus the corresponding maximum velocity separation. The dashed line is
the linear correlation best fit
Figure 13: Amplitude of the clustering of C IV lines as a function of
the column density. The upper right point has been obtained for the
present enlarged sample of C IV lines.
The lower left point has been derived from the data of the QSO 1422+231
(Songaila & Cowie 1996).
The vertical bars represent 
poissonian uncertainties in the
determination of the TPCF while the horizontal ones show the
confidence interval of the column densities