  Stochastic point processes with Coulomb interactions arise in various natural
examples of statistical mechanics, random matrices and optimization problems.
Often such systems due to their natural repulsion exhibit remarkable
hyperuniformity properties, that is, the number of points landing in any given
region 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 of
Jancovici, Lebowitz, Manificat, Martin, and Yalcin ('80,'83,'93), states that
the variance of the number of points landing in a set should grow like the
surface area instead of the volume unlike i.i.d. random points. In a recent
beautiful work, Chatterjee (2017) gave the first proof of such a result in
dimension three for a Coulomb type system, known as the hierarchical Coulomb
gas, inspired by Dyson's hierarchical model of the Ising ferromagnet. However
the case of dimensions greater than three had remained open. In this paper, we
establish the correct fluctuation behavior up to logarithmic factors in all
dimensions greater than three, for the hierarchical model. Using similar
methods, we also prove sharp variance bounds for smooth linear statistics which
were unknown in any dimension bigger than two. A key intermediate step is to
obtain precise results about the ground states of such models whose behavior
can be interpreted as hierarchical analogues of various crystalline conjectures
predicted for energy minimizing systems, and could be of independent interest.
