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. Typically, algorithms for computing the NPCF components have O(nN ) complexity (for a dataset 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 complexityO(n2) orO(ng log ng) if evaluated using a Fast Fourier Transform on a grid of size ng. 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 Julia package implementing our estimators and show how they can be applied to a variety of scenarios within cosmology and fluid dynamics.Theefficiency of such estimators will allowhigher-order correlators to become a standard tool in the analysis of random fields.