We investigate lower asymptotic bounds of number variances for invariant locally square-integrable random measures on hyperbolic spaces such as the unit disk. In particular, we prove that random measures are never geometrically hyperuniform and if the random measure admits non-trivial complementary series diffraction, then it is hyperfluctuating.
Reference: M. Björklund, M. Byléhn, "Hyperuniformity of random measures on Euclidean and hyperbolic spaces". Preprint.