Local density fluctuations, hyperuniformity, and order metrics

Questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges. The purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d. Our specific objectives are to study the variance in the number of points contained within a regularly shaped window Omega of arbitrary size, and to further illuminate our understanding of hyperuniform systems, i.e., point patterns that do not possess infinite-wavelength fluctuations. For large windows, hyperuniform systems are characterized by a local variance that grows only as the surface area…