Efficient computation of N -point correlation functions in D dimensions

Author:

Philcox Oliver H. E.12ORCID,Slepian Zachary34ORCID

Affiliation:

1. Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08540

2. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540

3. Department of Astronomy, University of Florida, Gainesville, FL 32611

4. Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94709

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 ( n N ) 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 complexity O ( n 2 ) or O ( n g log n g ) if evaluated using a Fast Fourier Transform on a grid of size n 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 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.

Funder

Simons Foundation

Publisher

Proceedings of the National Academy of Sciences

Subject

Multidisciplinary

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