Papers › Efficient Computation of N-point Correlation Functions in D Dimensions
Efficient Computation of N-point Correlation Functions in D Dimensions
Oliver H. E. Philcox, Zachary Slepian
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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 𝒪(nᴺ) 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 𝒪(n²), or 𝒪(n_glogn_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 \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.
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