Ground states and hyperuniformity of the hierarchical Coulomb gas in all dimensions

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 (see Jancovici et al. in J Stat Phys 72(3-4):773-787, 1993, Lebowitz in Phys Rev A 27:1491-1494, 1983, Martin and Yalcin in J Stat Phys 22(4):435-463, 1980), 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 in Rigidity of the three-dimensional hierarchical Coulomb gas, 2017. arXiv preprint., To appear in Probability Theory and Related Fields), Chatterjee 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 (Dyson in Phys Rev 92:1331-1338, 1953, Dyson in Commun Math Phys 12(2):91-107, 1969). 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.