# Computing 3 point correlation function randoms counts without the randoms catalogue

## TL;DR

This paper introduces a fast, novel method for calculating three-point correlation function randoms counts directly from data, significantly reducing computational time without needing random catalogues.

## Contribution

The authors develop a new approach that computes triplet counts using one-dimensional integrals, bypassing the traditional need for dense random catalogues in three-point statistics.

## Key findings

- Speeds up three-point function calculations by orders of magnitude.
- Eliminates the need for generating and counting random catalogues.
- Maintains accuracy while reducing computational resources.

## Abstract

As we move towards future galaxy surveys, the three-point statistics will be increasingly leveraged to enhance the constraining power of the data on cosmological parameters. An essential part of the three-point function estimation is performing triplet counts of synthetic data points in random catalogues. Since triplet counting algorithms scale at best as $\mathcal{O}(N^2\log N)$ with the number of particles and the random catalogues are typically at least 50 times denser than the data; this tends to be by far the most time-consuming part of the measurements. Here we present a simple method of computing the necessary triplet counts involving uniform random distributions through simple one-dimensional integrals. The method speeds up the computation of the three-point function by orders of magnitude, eliminating the need for random catalogues, with the simultaneous pair and triplet counting of the data points alone being sufficient.