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Disordered many-particle hyperuniform systems are exotic amorphous states ofmatter that lie between crystals and liquids. Hyperuniform systems haveattracted recent attention because they are endowed with novel transport andoptical properties. Recently, the hyperuniformity concept has been generalizedto characterize scalar fields, two-phase media and random vector fields. Inthis paper, we devise methods to explicitly construct hyperuniform scalarfields. We investigate explicitly spatial patterns generated from Gaussianrandom fields, which have been used to model the microwave background radiationand heterogeneous materials, the Cahn-Hilliard equation for spinodaldecomposition, and Swift-Hohenberg equations that have been used to modelemergent pattern formation, including Rayleigh-B{\' e}nard convection. We showthat the Gaussian random scalar fields can be constructed to be hyperuniform.We also numerically study the time evolution of spinodal decomposition patternsand demonstrate that these patterns are hyperuniform in the scaling regime.Moreover, we find that labyrinth-like patterns generated by the Swift-Hohenbergequation are effectively hyperuniform. We show that thresholding a hyperuniformGaussian random field to produce a two-phase random medium tends to destroy thehyperuniformity of the progenitor scalar field. We then propose guidelines toachieve effectively hyperuniform two-phase media derived from thresholdednon-Gaussian fields. Our investigation paves the way for new researchdirections to characterize the large-structure spatial patterns that arise inphysics, chemistry, biology and ecology. Moreover, our theoretical results areexpected to guide experimentalists to synthesize new classes of hyperuniformmaterials with novel physical properties via coarsening processes and usingstate-of-the-art techniques, such as stereolithography and 3D printing.

Random Scalar Fields and Hyperuniformity