Hyperuniformity in regular trees

TL;DR

This paper investigates hyperuniformity properties of point processes on regular trees, revealing limitations and providing examples of stealthy configurations with sub-volume variance growth.

Contribution

It introduces spectral hyperuniformity and stealth concepts for point processes on trees and constructs novel stealthy lattice orbits with sub-volume variance growth.

Findings

Regular tree point processes are not geometrically hyperuniform.

Processes with diffraction support in the complementary series are hyperfluctuating.

Constructed stealthy lattice orbits with variance growth slower than volume.

Abstract

We study notions of hyperuniformity for invariant locally square-integrable point processes in regular trees. We show that such point processes are never geometrically hyperuniform, and if the diffraction measure has support in the complementary series then the process is geometrically hyperfluctuating along all subsequences of radii. A definition of spectral hyperuniformity and stealth of a point process is given in terms of vanishing of the complementary series diffraction and sub-Poissonian decay of the principal series diffraction near the endpoints of the principal spectrum. Our main contribution is providing examples of stealthy invariant random lattice orbits in trees whose number variance grows strictly slower than the volume along some unbounded sequence of radii. These random lattice orbits are constructed from the fundamental groups of complete graphs and the Petersen graph.