Tilings of space and superhomogeneous point processes

TL;DR

This paper presents methods to construct superhomogeneous point processes from tilings of Euclidean space, showing how to control their structure factor behavior at small wavenumbers and providing explicit examples with potential applications in physics and cosmology.

Contribution

It introduces a novel analytical framework for generating superhomogeneous point processes from tilings, including explicit constructions with high structure factor exponents.

Findings

Point processes with vanishing structure factor at zero wavenumber can be constructed from tilings.

Assigning points to tile centers yields a b3=4 exponent for short-range correlated tiles.

Long-range correlations and multiple points per tile allow higher b3 exponents, exceeding 4.

Abstract

We consider the construction of point processes from tilings, with equal volume tiles, of d-dimensional Euclidean space. We show that one can generate, with simple algorithms ascribing one or more points to each tile, point processes which are "superhomogeneous'' (or "hyperuniform''), i.e., for which the structure factor S(k) vanishes when the wavenumber k tends to zero. The exponent of the leading small-k behavior depends in a simple manner on the nature of the correlation properties of the specific tiling and on the conservation of the mass moments of the tiles. Assigning one point to the center of mass of each tile gives the exponent \gamma=4 for any tiling in which the shapes and orientations of the tiles are short-range correlated. Smaller exponents, in the range 4-d<\gamma<4 (and thus always superhomogeneous for d\leq 4), may be obtained in the case that the latter quantities have long-range correlations. Assigning more than one point to each tile in an appropriate way, we show that one can obtain arbitrarily higher exponents in both cases. We illustrate our results with explicit constructions using known deterministic tilings, as well as some simple stochastic tilings for which we can calculate S(k) exactly. Our results provide, we believe, the first explicit analytical construction of point processes with \gamma > 4. Applications to condensed matter physics, and also to cosmology, are briefly discussed.