Generalized cumulant correlators and hierarchical clustering

The cumulant correlators, C-pq, are statistical quantities that generalize the better-known S-p parameters; the former are obtained from the two-point probability distribution function of the density fluctuations while the la tter describe only the one-point distribution. If galaxy clustering develop s from Gaussian initial fluctuations and a small-angle approximation is ado pted, standard perturbative methods suggest a particular hierarchical relat ionship of the C-pq for projected clustering data, such as that obtained fr om the Automatic Plate Measuring (APM) survey. We establish the usefulness of the two-point cumulants for describing hierarchical clustering by compar ing such calculations against available measurements from projected catalog ues, finding very good agreement. We extend the idea of cumulant correlator s to multipoint generalized cumulant correlators (related to the higher-ord er correlation functions). We extend previous studies in the highly non-lin ear regime to express the generalized cumulant correlators in terms of the underlying 'tree amplitudes' of hierarchical scaling models. Such considera tions lead to a technique for determining these hierarchical amplitudes, to arbitrary order, from galaxy catalogues and numerical simulations. Knowled ge of these amplitudes yields important clues about the phenomenology of gr avitational clustering. For instance, we show that a three-point cumulant c orrelator can be used to separate the tree amplitudes up to sixth order. We also combine the particular hierarchical Ansatz of Bernardeau & Schaeffer with extended and hyper-extended perturbation theory to infer values of the tree amplitudes in the highly non-linear regime.