Efficient computation of N-point correlation functions in D dimensions

Area of Science:

• Physics
• Cosmology
• Fluid Dynamics
• Data Analysis

Background:

• N-point correlation functions (NPCFs) are crucial for describing random fields in physical sciences.
• Existing algorithms for NPCFs have high computational complexity, limiting their application to small N.
• Efficient computation of NPCFs is needed for advanced analysis of stochastic processes.

Purpose of the Study:

• To develop efficient algorithms for computing NPCFs in D-dimensional spaces.
• To reduce the computational complexity of NPCF calculations.
• To enable the use of higher-order correlation functions as a standard analytical tool.

Main Methods:

• Projecting NPCFs onto a D-dimensional hyperspherical harmonic basis.
• Utilizing a separable form for estimators.
• Employing Fast Fourier Transform for grid-based evaluation.
• Implementing algorithms in a Julia package.

Main Results:

• Achieved computational complexity of O(n^2 log n) or O(n log n) using FFT.
• Demonstrated significant dimensionality reduction for isotropic correlation functions.
• Developed a practical Julia package for NPCF estimation.

Conclusions:

• The new algorithms offer a computationally feasible method for calculating NPCFs.
• This work facilitates the use of higher-order correlation functions in diverse scientific fields.
• The developed methods will advance the analysis of random fields in cosmology and fluid dynamics.