{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/efficient-computation-of-n-point-correlation","title":"Efficient Computation of $N$-point Correlation Functions in $D$ Dimensions","arxiv_id":"2106.10278","date":"2021-06-18","proceeding":null,"authors":["Oliver H. E. Philcox","Zachary Slepian"],"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.","url_abs":"https://arxiv.org/abs/2106.10278v2","url_pdf":"https://arxiv.org/pdf/2106.10278v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"efficient-computation-of-n-point-correlation","repo_url":"https://github.com/oliverphilcox/encore","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"efficient-computation-of-n-point-correlation","repo_url":"https://github.com/oliverphilcox/npcfs.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}