Hyperuniform and rigid stable matchings

With respect to the degree of order and disorder, a snapshot of the ideal gas with completely independent particle positions (in mathematics known as the realization of a Poisson point process) is the exact opposite of a lattice with a perfect short- and long-range order.

We have defined a stable matching between these two extremes and proved that the resulting pattern can locally be virtually indistinguishable from the ideal gas but at large scales it inherits two remarkable global properties of the lattice: hyperuniformity and number rigidity, as demonstrated in our supplementary video.

M. A. Klatt, G. Last, D. Yogeshwaran. Hyperuniform and rigid stable matchings. Random Struct Alg. 57:439–473 (2020)

In the stable matching, points of the lattice and the ideal gas prefer to be close to each other. Assuming that there are on average more points per unit area in the ideal gas than in the lattice, the matched points in the ideal gas form a new point process that inherits properties from both the lattice and the ideal gas.

In fact, if the mean number of points per unit area converge, the new point pattern becomes almost indistinguishable from the ideal gas in any finite observation window, while its large-scale density fluctuations remain anomalously suppressed similar to the lattice.

We prove that the resulting point process is hyperuniform and number rigidity also for a stable matching between a lattice and not only the ideal gas but a class of correlated point processes, known as determinantal point processes. For a one-sided matching in 1D, our proof applies to quite general point processes.

Our paper solves an open question from the seminal paper by Salvatore Torquato and Frank Stillinger that introduced the notion of hyperuniformity. Our matched point processes generalizes a 1D hyperuniform point process by Goldstein, Lebowitz, Speer.

You can easily match two arbitrary point patterns using my R-package: matchingpp