Generalised Cumulant Correlators and Hierarchical Clustering

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.