  The cumulant correlators, $C_{pq}$, are statistical quantities that
generalise the better-known $S_p$ parameters; the former are obtained from the
two-point probability distribution function of the density fluctuations while
the latter describe only the one-point distribution. If galaxy clustering
develops from Gaussian initial fluctuations and a small-angle approximation is
adopted, standard perturbative methods suggest a particular hierarchical
relationship of the $C_{pq}$ for projected clustering data, such as the APM
survey. We establish the usefulness of the two-point cumulants for describing
hierarchical clustering by comparing such calculations against available
measurements from projected catalogs, finding very good agreement. We extend
the idea of cumulant correlators to multi-point {\em generalised} cumulant
correlators (related to the higher-order correlation functions). We extend
previous studies in the highly nonlinear regime to express the generalised
cumulant correlators in terms of the underlying ``tree amplitudes'' of
hierarchical scaling models. Such considerations lead to a technique for
determining these hierarchical amplitudes, to arbitrary order, from galaxy
catalogs and numerical simulations. Knowledge of these amplitudes yields
important clues about the phenomenology of gravitational clustering. For
instance, we show that three-point cumulant correlator can be used to separate
the tree amplitudes up to sixth order. We also combine the particular
hierarchical {\em ansatz} of Bernardeau & Schaeffer (1992) with extended and
hyperextended perturbation theory to infer values of the tree amplitudes in the
highly nonlinear regime.
