Large ensembles of points with Coulomb interactions arise in various settings 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 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 from the eighties, 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 2017, 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. However the case of dimensions greater than three had remained open.

We will discuss a recent result establishing the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three, for the hierarchical model. The proof relies on a precise understanding of ground states of such models and using this as an input along with a technique reminiscent of the cavity method from the theory of spin glasses to obtain sharp estimates of the partition function. 

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