Stable Matching of Point Processes and Hyperuniformity

Abstract

Hyperuniformity is a global property of stationary point processes. The variance of the number of points of such processes within an observation window grows substantially more slowly than the volume of the observation window. At a macroscopic level, this results in rather uniformly distributed points. This distinguishes hyperuniform point processes from, for example, Poisson processes, where points are not correlated, and the variance is proportional to the volume of the observation window. The talk explores the construction of hyperuniform thinnings of a broad class of mixing stationary point processes. For a small thinning parameter, the thinned point process is locally hardly distinguishable from the original point process. The construction is based on stable matching, which describes a specific pairwise matching of points from two point processes, favoring points in close proximity. The concept is known from the "stable marriage problem", which in this context is uniquely solvable by a simple algorithm. Here the target point process is stably matched with a hyperuniform point process of lower intensity. Upon disregarding points of the target process that lack a matching partner, the resulting thinning inherits hyperuniformity. This work extends a prior result in [1], where a more limited class of point processes was considered. [1] Klatt, M., Last, G., and Yogeshwaran, D. (2018). Hyperuniform and Rigid Stable Matchings.