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\def\xiav{\avrg{\xi}} \def\xiN{\avrg{\xi^{(N)}}} \def\avrg#1{\langle{#1} \rangle}
We study the higher order correlation properties of the distribution of galaxies in the Lick, IRAS, and APM catalogs. We show, using a new technique, how the much debated cluster correlation function is directly related to the higher order galaxy correlations. Mathematical techniques from biased galaxy formation are adopted: here the biasing simulates the detection of the clusters, rather than their dynamical formation. For statistical purposes we define the factorial moment correlators as $$ w_{kl}(r_{12}) = { \avrg{(N_1)_k (N_2)_l}-\avrg{(N)_k} \avrg{(N)_l} \over\avrg{N}^{k+l} }, $$ where we used the notatation $(N)_k = N(N-1)..(N-1+k)$ for the factorial moments. Since these correspond to the continuum moments after the transition to discrete we essentially calculate the correlation functions in a catalog in which the density is raised to a power. This is a `soft clipping' because it enhances the density contrast similarly to `sharp clipping', where clusters are identified as cells with density exceeding a certain threshold. The advantage of `soft clipping' is that the $w_{kl}$'s are obtained as a finite sum of higher order correlation functions. The resulting `cluster' catalogs have amplifications similar to rich clusters, and the galaxy-`cluster' cross-correlation is close to the geometric mean of the galaxy and cluster autocorrelations. We also present a novel method to determine the amplitudes of the irreducible N-point correlation functions in the gravitational clustering hierarchy. The case of `sharp clipping' is investigated theoretically with the method of generating functions. It is shown that the shape of the cluster correlation function is always identical to the galaxy correlation function and we calculate the amplification for some special cases. The theoretical results agree well with known observations.
