Hyperuniformity of random measures on Euclidean and hyperbolic spaces

TL;DR

This paper explores the behavior of number variances and hyperuniformity properties of invariant random measures on Euclidean and hyperbolic spaces, revealing fundamental differences and new spectral concepts.

Contribution

It establishes lower bounds for number variances, shows non-hyperuniformity in hyperbolic spaces, and introduces spectral hyperuniformity and stealth in this context.

Findings

Number variance grows at least as fast as boundary volume in Euclidean spaces.

Random measures in hyperbolic spaces are never geometrically hyperuniform.

Presence of non-trivial diffraction implies hyperfluctuating measures.

Abstract

We investigate lower asymptotic bounds of number variances for invariant locally square-integrable random measures on Euclidean and real hyperbolic spaces. In the Euclidean case we show that there are subsequences of radii for which the number variance grows at least as fast as the volume of the boundary of Euclidean balls, generalizing a classical result of Beck. With regards to real hyperbolic spaces we prove that random measures are never geometrically hyperuniform and if the random measure admits non-trivial complementary series diffraction, then it is hyperfluctuating. Moreover, we define spectral hyperuniformity and stealth of random measures on real hyperbolic spaces in terms of vanishing of the complementary series diffraction and sub-Poissonian decay of the principal series diffraction around the Harish-Chandra $\Xi$-function.