Hyperuniform and rigid stable matchings

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.