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We consider the construction of point processes from tilings, with equalvolume tiles, of d-dimensional Euclidean space. We show that one can generate,with simple algorithms ascribing one or more points to each tile, pointprocesses which are "superhomogeneous'' (or "hyperuniform''), i.e., for whichthe structure factor S(k) vanishes when the wavenumber k tends to zero. Theexponent of the leading small-k behavior depends in a simple manner on thenature of the correlation properties of the specific tiling and on theconservation of the mass moments of the tiles. Assigning one point to thecenter of mass of each tile gives the exponent \gamma=4 for any tiling in whichthe shapes and orientations of the tiles are short-range correlated. Smallerexponents, in the range 4-d<\gamma<4 (and thus always superhomogeneous ford\leq 4), may be obtained in the case that the latter quantities havelong-range correlations. Assigning more than one point to each tile in anappropriate way, we show that one can obtain arbitrarily higher exponents inboth cases. We illustrate our results with explicit constructions using knowndeterministic tilings, as well as some simple stochastic tilings for which wecan calculate S(k) exactly. Our results provide, we believe, the first explicitanalytical construction of point processes with \gamma > 4. Applications tocondensed matter physics, and also to cosmology, are briefly discussed.

Tilings of space and superhomogeneous point processes