On the Fast Random Sampling and Other Properties of the Three Point
  Correlation Function in Galaxy Surveys
  In the forthcoming large volume galaxy surveys higher order statistics will
provide complementary information to the usual two point statistics. Low
variance estimators of the Three Point Correlation Function (3CPF) of discrete
data count triangle configurations with vertices mixing data and random
catalogues. Large density random catalogues are used to reduce the shot noise,
which leads to a computational cost of one or two orders of magnitude more than
the pure data histogram. In this paper, we explore time reductions of the
isotropic 3PCF random sampling terms in periodic boxes without using random
catalogues. In the first approach, based on Hamilton's construction of his
famous two point estimator, we use an ad-hoc two point correlation term, while
for the second procedure we construct the operators from a geometrical
viewpoint, using two sides and their opening angle to describe the 3PCF
triangle configurations. We map the last result to the three triangle side
basis either numerically or analytically, and show that the latter approach
performs best when applied to synthetic data. Moreover, we elaborate on going
beyond periodic boxes, discuss other low variance n-point estimators and
present useful 3PCF visualization schemes.
