Efficient Computation of $N$-point Correlation Functions in $D$ Dimensions

Abstract

We present efficient algorithms for computing the $N$-point correlation functions (NPCFs) of random fields in arbitrary $D$-dimensional homogeneous and isotropic spaces. Such statistics appear throughout the physical sciences, and provide a natural tool to describe stochastic processes. algorithms for computing the NPCF components have $\mathcal{O}(n^N)$ complexity (for a data set containing $n$ particles); their application is thus computationally infeasible unless $N$ is small. By projecting the statistic onto a suitably-defined angular basis, we show that the estimators can be written in a separable form, with complexity $\mathcal{O}(n^2)$, or $\mathcal{O}(n_{\rm g}\log n_{\rm g})$ if evaluated using a Fast Fourier Transform on a grid of size $n_{\rm g}$. Our decomposition is built upon the $D$-dimensional hyperspherical harmonics; these form a complete basis on the $(D-1)$-sphere and are intrinsically related to angular momentum operators. Concatenation of $(N-1)$ such harmonics gives states of definite combined angular momentum, forming a natural separable basis for the NPCF. As $N$ and $D$ grow, the number of basis components quickly becomes large, providing a practical limitation to this (and all other) approaches: however, the dimensionality is greatly reduced in the presence of symmetries; for example, isotropic correlation functions require only states of zero combined angular momentum. We provide a \textsc{Julia} package implementing our estimators, and show how they can be applied to a variety of scenarios within cosmology and fluid dynamics. The efficiency of such estimators will allow higher-order correlators to become a standard tool in the analysis of random fields.