On estimating the structure factor of a point process, with applications to hyperuniformity

Hyperuniformity is the study of stationary point processes with a sub-Poisson variance of the number of points in a large window. In other words, counting the points of a hyperuniform point process that fall in a given large region yields a small-variance Monte Carlo estimator of the volume of that region. Hyperuniform point processes have received a lot of attention in statistical physics, both for the investigation of natural organized structures and the synthesis of materials. Unfortunately, rigorously proving that a point process is hyperuniform is usually difficult. A common practice in statistical physics and chemistry is to use a few point process samples to estimate a spectral measure called the structure factor. Plotting the estimated structure factor, and evaluating its decay around zero, provides a graphical diagnostic of hyperuniformity. Different applied fields use however different estimators, and important algorithmic choices proceed from each field's lore. This paper is a systematic survey and derivation of known or otherwise natural estimators of the structure factor. In an effort to make investigations of the structure factor and hyperuniformity systematic and reproducible, we further provide a Python toolbox strucutre_factor, containing all the estimators and tools that we discuss.