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Stochastic point processes with Coulomb interactions arise in various naturalexamples of statistical mechanics, random matrices and optimization problems.Often such systems due to their natural repulsion exhibit remarkablehyperuniformity properties, that is, the number of points landing in any givenregion fluctuates at a much smaller scale compared to that of a set of i.i.d.random points. A well known conjecture from physics appearing in the works ofJancovici, Lebowitz, Manificat, Martin, and Yalcin ('80,'83,'93), states thatthe variance of the number of points landing in a set should grow like thesurface area instead of the volume unlike i.i.d. random points. In a recentbeautiful work, Chatterjee (2017) gave the first proof of such a result indimension three for a Coulomb type system, known as the hierarchical Coulombgas, inspired by Dyson's hierarchical model of the Ising ferromagnet. Howeverthe case of dimensions greater than three had remained open. In this paper, weestablish the correct fluctuation behavior up to logarithmic factors in alldimensions greater than three, for the hierarchical model. Using similarmethods, we also prove sharp variance bounds for smooth linear statistics whichwere unknown in any dimension bigger than two. A key intermediate step is toobtain precise results about the ground states of such models whose behaviorcan be interpreted as hierarchical analogues of various crystalline conjecturespredicted for energy minimizing systems, and could be of independent interest.

Ground States And Hyperuniformity Of The Hierarchical Coulomb Gas In All Dimensions