Tilings of space and superhomogeneous point processes

Abstract

We consider the construction of point processes from tilings, with equal-volume tiles, of d -dimensional Euclidean space R;{d} . 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 wave vector k tends to zero. The exponent gamma characterizing the leading small- k behavior, S(k-->0) proportional, variant k(gamma), 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< or =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 an explicit analytical construction of point processes with gamma>4 . Applications to condensed matter physics, and also to cosmology, are briefly discussed.