Hyperuniform and rigid stable matchings

TL;DR

This paper investigates the properties of stable matchings between lattice points and stationary determinantal point processes, revealing hyperuniformity and rigidity in the resulting matched point processes, contrasting with Poisson processes.

Contribution

It introduces a general framework for analyzing stable matchings with subexponential tail conditions, establishing hyperuniformity and rigidity properties of the matched process.

Findings

Matched points form a hyperuniform and number rigid process

The matched process has an exponentially decreasing pair correlation function if the original is Poisson

Results apply to processes with certain tail and mixing conditions

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.