Rigidity of the three-dimensional hierarchical Coulomb gas

Abstract

A random set of points in Euclidean space is called 'rigid' or 'hyperuniform' if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i.i.d. random points. This phenomenon has received considerable attention in recent years, due to its appearance in random matrix theory, the theory of Coulomb gases and zeros of random analytic functions. However, most of the published results are in dimensions one and two. This paper gives the first proof of hyperuniformity in a Coulomb type system in dimension three, known as the hierarchical Coulomb gas. This is a simplified version of the actual 3D Coulomb gas. The interaction potential in this model, inspired by Dyson's hierarchical model of the Ising ferromagnet, has a hierarchical structure and is locally an approximation of the Coulomb potential. Hyperuniformity is proved at both macroscopic and microscopic scales, with upper and lower bounds for the order of fluctuations that match up to logarithmic factors. The fluctuations have cube-root behavior, in agreement with a well-known prediction for the 3D Coulomb gas. For completeness, analogous results are also proved for the 2D hierarchical Coulomb gas and the 1D hierarchical log gas.

Quadrature Points via Heat Kernel Repulsion

“…In the case of the "dented" sphere example, we initialize x (k) by mapping the spherical Fibonacci point set (for N = 89) onto the "dented" sphere using the projection map (x 1 , x 2 , x 3 ) → (x 1 , sign(x 2 ) (α + x 2 1 )x 2 2 , x 3 ). In the example of the Poincaré disk model we initialize the particles by first generating uniformly distributed points on the disk {(x 1 , x 2 ) : x 2 1 + x 2 2 ≤ 4/5}, and then project these points onto the upper sheet of the hyperboloid model using the appropriately defined inverse of the projection (9). During simulation time the additional constraint (10) for the compactified hyperboloid is ensured to be (approximately) satisfied by adding the additional energy term Ũ (x) = N i=1 Ũi (x i ), to the energy functional E Gaussian , where Ũi ((x i,1 , x i,2 , x i,3 )) = 0, if |x i,3 | ≤ c, κ(x i,3 − c) α , otherwise with c = 1+r 2 1−r 2 , r = 4/5, and sufficiently large α > 1, κ > 0.…”

“…The construction of support points in this way is illustrated in Figure 10 (A), which shows a point set of N = 150 points on the compactified upper sheet of the hyperboloid constructed as a minimum energy configuration of the energy functional E Gaussian . The image of this point set under the projection (9), is shown in Figure 10 (B). As explained above this point set corresponds by construction to a minimum energy configuration of the energy functional E Gaussian on the compactified Poincaré disk model.…”

“…The second question is also of relevance in mathematical physics (see e.g. [1,9,14,46], we refer to a survey of Blanc & Lewin on the Crystallization Conjecture [3]).…”

Hyperuniform and rigid stable matchings

“…The ones we are aware of are small i.i.d. Gaussian perturbations of a lattice [31], stationary point processes satisfying DLR (Dobrushin-Landford-Ruelle) equations with appropriate interacting potentials [7] and the hierarchical Coulomb gas in d = 3 [5]. We wish to also point out that first rigidity and hyperuniformity reveal something intrinsically interesting about the point process, and second these properties are also useful to understand percolation models on point processes (see page 5 of [13]).…”

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

“…random points. In a recent beautiful work [18], 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 [22,23]. However the case of dimensions greater than three had remained open.…”