  We examine the correlational property of $\lya$ clouds in detail and compare
it to that of mass and galaxies. We show that the correlation strength of
$\lya$ clouds is somewhat weaker than that of the underlying matter, which in
turn is weaker than that of galaxies (biased galaxy formation). On the scales
probed, $10-300$km/s, for $\lya$ clouds, we find that higher density, higher
optical depth, higher column density regions are more strongly clustered than
lower density, lower optical depth, lower column density regions. Thus, a
consistent picture seems to emerge: the correlation strength for a given set of
objects is positively correlated with their characteristic global density and
the differences among the correlations of galaxies, $\lya$ clouds and mass
reflect the differences in density that each trace. Significant positive
correlations with a strength of $0.1-1.0$ are found for $\lya$ clouds in the
velocity range $50-300$km/s. Among the correlational measures examined, an
optical depth correlation function (Equation 5) proposed here may serve as the
best correlational measure. It reasonably faithfully represents the true
correlation of the underlying matter, enabling a better indication of both
matter correlation and the relationship between galaxies and $\lya$ clouds.
Furthermore, it appears to be an alternative to the conventional line-line
correlation function with the virtue that it does not require ambiguous
post-observation fitting procedures such as those commonly employed in the
conventional line-finding methods. Neither does it depend sensitively on the
observational resolution (e.g., FWHM), insofaras the clouds are resolved
Conveniently, it can be easily measured with the current observational
sensitivity without being contaminated significantly by the presence of noise.

Astrophysics