  This work presents a set of new statistics, the cumulant correlators, aimed
at high precision analysis of the galaxy distribution. They form a symmetric
matrix, $Q_{NM}$, related to moment correlators the same way as cumulants are
related to the moments of the distribution. They encode more information than
the usual cumulants, $S_N$'s, and their extraction from data is similar to the
calculation of the two-point correlation function. Perturbation theory (PT),
its generalization, the extended perturbation theory (EPT), and the
hierarchical assumption (HA) have simple predictions for these statistics. As
an example, the factorial moment correlators measured by Szapudi, Dalton,
Efstathiou & Szalay (1996) in the APM catalog are reanalyzed using this
technique. While the previous analysis assumed hierarchical structure
constants, this method can directly investigate the validity of HA, along with
PT, and EPT. The results in agreement with previous findings indicate that, at
the small scales used for this analysis, the APM data supports HA. When all
non-linear corrections are taken into account it is a good approximation at the
20 percent level. It appears that PT, and a natural generalization of EPT for
cumulant correlators does not provide such a good fit for the APM at small
scales. Once the validity the HA is approximately established, cumulant
correlators can separate the amplitudes of different tree-types in the
hierarchy up to fifth order. As an example, the weights for the fourth order
tree topologies are calculated including all non-linear corrections.
