Hyperuniform and rigid stable matchings

Abstract

We study a stable partial matching τ of the d‐dimensional lattice with a stationary determinantal point process Ψ on Rd with intensity α>1. For instance, Ψ might be a Poisson process. The matched points from Ψ form a stationary and ergodic (under lattice shifts) point process Ψτ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψτ 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 Ψ, whose so‐called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behavior. For hyperuniformity, we also additionally need to assume some mixing condition on Ψ. Furthermore, if Ψ is a Poisson process then Ψτ has an exponentially decreasing truncated pair correlation function.