Weighted Point Configurations with Hyperuniformity: An Ecological Example and Models

Random point configurations are said to be in hyperuniform states, if density fluctuations are anomalously suppressed in large-scale. Typical examples are found in Coulomb gas systems in two dimensions especially called log-gases in random matrix theory, in which points are repulsively correlated by long-range potentials. In infertile lands like deserts continuous survival competitions for water and nutrition will cause long-ranged repulsive interactions among plants. We have prepared digital data of spatial configurations of center-of-masses for bushes weighted by bush sizes which we call masses. Data analysis shows that such ecological point configurations do not show hyperuniformity as unmarked point processes, but are in hyperuniform states as marked point processes in which mass distributions are taken into account. We propose the non-equilibrium statistical-mechanics models to generate marked point processes having hyperuniformity, in which iterations of random thinning of points and coalescing of masses transform initial uncorrelated point processes into non-trivial point processes with hyperuniformity. Combination of data analysis and computer simulations shows the importance of strong correlations in probability law between spatial point configurations and mass distributions of individual points to realize hyperuniform marked point processes.