On the fast random sampling and other properties of the three point correlation function in galaxy surveys

In the forthcoming large volume galaxy surveys, the use of higher order statistics will prove important to obtain complementary information to the usual two point statistics. In particular, one of those higher order statistical tools is the Three Point Correlation Function (3PCF) over discrete data points, whose lowest variance estimators count triangle configurations with vertices mixing data and random catalogues. A popular choice is to use large density random catalogues, which reduces the shot noise but leads to a computational cost of one or two orders of magnitude more than the pure data histogram calculation. In this paper, we explore ideas to time reduce the random sampling without using random catalogues. We focus on the isotropic 3PCF case over periodic boxes. In a first approach, based on Hamilton's construction of his famous two point estimator, we use an ad-hoc two point correlation piece for the mixed random-data histograms. A second approach relies on operators constructed from a geometrical viewpoint, using two sides and one angle to define the three dependencies of the isotropic 3PCF . We map the last result to the three triangle side basis either numerically or analytically, and show that the latter method performs best when applied to synthetic data. In addition, we elaborate on relaxing the no boundary condition, discuss other low variance n-point estimators and present useful 3PCF visualization schemes.