Hyperuniformity in regular trees

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.