A Count Probability Cookbook: Spurious Effects and the Scaling Model

We study the errors brought by finite volume effects and dilution effects on the practical determination of the count probability distribution function P_N(n,L), which is the probability of having N objects in a cell of volume L^3 for a set of average number density n. Dilution effects are particularly relevant to the so-called sparse sampling strategy. This work is mainly done in the framework of the scaling model (Balian \& Schaeffer 1989), which assumes that the Q-body correlation functions obey the scaling relation xi_Q(K r_1,..., K r_Q) = K^{-(Q-1) gamma} xi_N(r_1,..., r_Q). We use three synthetic samples as references to perform our analysis: a fractal generated by a Rayleigh-L\'evy random walk with 3.10^4 objects, a sample dominated by a spherical power-law cluster with 3.10^4 objects and a cold dark matter (CDM) universe involving 3.10^5 matter particles.