**A. Higher order correlation functions**

The two-point correlation function is not a unique descriptor of
clustering, it is merely the first of an infinite hierarchy of
such descriptors describing the galaxy distribution of galaxies
taken *N* at a time. Two quite different distributions can have
the same two-point correlation function. In particular, the fact
that a point distribution generated by any random walk (e.g., as a
Lévy flight as proposed by
Mandlebrot (1975)
has the correct two-point correlation function does not mean much unless
other statistical measures of clustering are tested.

The present day galaxy distribution is manifestly not a Gaussian random process: there is, for example, no symmetry about the mean density. This fact alone tells us that there is more to galaxy clustering than the two-point correlation function.

So what kind of descriptors should we look for? Generalizations of
the two-point functions to 3-, 4- and higher order functions are
certainly possible, but they are difficult to calculate and not
particularly edifying. However, they do the job of providing some
of the needed extra information and through such constructs as the
BBGKY hierarchy ^{9} they do
relate to the underlying physics of the
clustering process. We shall describe the observed scaling of the
3-point correlation function below.

One alternative is to go for different clustering models: anything but correlation functions. These may have the virtue of providing immediate gratification in terms of visualization of the process, but they are often difficult to relate to any kind of dynamical process.

If we knew all higher order correlation functions we would have a
complete description of the galaxy clustering process. However,
calculating an estimate of a two point function from a sample of
*N* galaxies requires taking all pairs from the sample of *N*,
while calculating a three point functions requires taking all
triples from *N*. The amount of computation escalates rapidly and
restrictions have to be imposed on what is actually being calculated.

Nevertheless, calculating restricted *N*-point functions may be
useful: these functions may be related to one another and have
interesting scale dependence.
Gaztañaga (1992)
has calculated restricted *N*-point functions and showed that these
have power law behavior over the range of scales where they can be
determined.

^{9} The BBGKY hierarchy, (after Bogolyubov,
Born, Green, Kirkwood and Yvon),
is an infinite chain of equations adapted from plasma physics
(Ichimaru, 1992)
to describe self-gravitating non-linear clustering (see for example
Fall and Severne
(1976),
Peebles (1980), and
Saslaw (2000).)
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