Hyperuniform and rigid stable matchings

Abstract

We study a stable partial matching τ of the (possibly randomized) d-dimensional lattice with a stationary determinantal point process Ψ on R d 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 Ψ τ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψ τ 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 behaviour. For hyperuniformity, we also additionally need to assume some mixing condition on Ψ. Further, if Ψ is a Poisson process then Ψ τ has an exponentially decreasing truncated pair correlation function.

Cloaking the underlying long-range order of randomly perturbed lattices

“…A perturbed lattice is a point pattern (process) in d-dimensional Euclidean space R d obtained by displacing each point in a Bravais lattice [5] according to some stochastic rule [1,[6][7][8]. Perturbed lattices have been intensively studied in a broad range of contexts, from statistical physics and cosmology [9,10] to crystallography lattices [1,2] or to probability theory, including distributions of zeros of random entire functions [11] and number rigidity [12][13][14]. They are related to certain queueing problems [15], in particular, G processes [16], and stable matchings in any dimension [14].…”

“…Perturbed lattices have been intensively studied in a broad range of contexts, from statistical physics and cosmology [9,10] to crystallography lattices [1,2] or to probability theory, including distributions of zeros of random entire functions [11] and number rigidity [12][13][14]. They are related to certain queueing problems [15], in particular, G processes [16], and stable matchings in any dimension [14]. Perturbed lattices are moreover used to generate disordered initial configurations for numerical simulations [17] or configurations of sampling points [18].…”

“…If both exist, then the perturbed lattice is class I hyperuniform with σ 2 (R) ∼ R d−1 , that is, the number variance grows like the surface area of the observation window. Further examples of class I hyperuniform systems are all crystals [19], many quasicrystals [32], certain random organization models [33], certain non-equilibrium dynamic states with active particles [34], some stable matchings [14], one-component plasmas [35,36], the Ginibre process related to random matrices [36][37][38], and hyperuniform disordered ground states [39,40]. The latter have been found particularly useful for optical applications, including photonic band gap materials [41], light extraction [42,43], and transparent low-density amorphous materials [44].…”

“…It measures deviations of two-point statistics (i.e., structure factor and pair correlation function) from that of the ideal gas (Poisson point process): In what follows, we will compute both Λ and τ to thoroughly characterize the degree of order and disorder in hyperuniform perturbed lattices. Currently, perturbed lattices with weak or no correlations are among the rare examples of amorphous hyperuniform point patterns that can be easily simulated with a million particles per sample [8,14,52,53]. However, in general, the resulting point patterns are not fully amorphous in the sense that their structure factor exhibits Bragg peaks, which are 'inherited' from the original lattice.…”