Hyperuniform and rigid stable matchings

Abstract

We study a stable partial matching $\tau$ of the (possibly randomized) $d$-dimensional lattice with a stationary determinantal point process $\Psi$ on ${\mathbb{R}}^d$ with intensity $\alpha >1$. For instance, $\Psi$ might be a Poisson process. The matched points from $\Psi$ form a stationary and ergodic (under lattice shifts) point process ${\Psi }^{\tau }$ with intensity $1$ that very much resembles $\Psi$ for $\alpha$ close to $1$. On the other hand ${\Psi }^{\tau }$ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process $\Psi$, whose so-called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behaviour. For hyperuniformity, we also additionally need to assume some mixing condition on $\Psi$. Further, if $\Psi$ is a Poisson process then ${\Psi }^{\tau }$ has an exponentially decreasing truncated pair correlation function.

Pages: 35, Figures: 10, Supplementary Video: Zooming out of a Poisson point process and a hyperuniform and rigid stable matching, Supplementary Animations: Visualization of the mutual nearest neighbor matching algorithm and the construction of a matching flower, Second version: Some typos and minor inaccuracies have been corrected