Maximal Rigidity of Random Measures and Uniqueness Pairs: Stealthy Processes, Quasicrystals and Periodicity

Abstract

This article investigates the phenomenon of maximal rigidity in random spatial systems, where perfect interpolation of the process is possible from partial information, specifically, from its restriction to a strict subdomain, often resulting in a trivial tail \(\sigma \)-algebra. A classical example known since the 1930’s is that a time series is deterministic, i.e. fully determined by its values on the negative integers if its spectrum has a gap, or at least a sufficiently deep zero. We extend such results to higher dimensions and continuous settings by establishing a connection with the concept of uniqueness pairs, rooted in the uncertainty principle of harmonic analysis. We present several other manifestations of this principle, unify and strengthen seemingly unrelated results across different models: stealthy processes are shown to be maximally rigid on cones, and for quasicrystals this cone can be arbitrarily small; discrete integer-valued processes are necessarily periodic when they have a simply connected spectrum. Finally, we identify a surprising class of continuous fields with seemingly standard behavior, such as linear variance and finite dependency range, that undergo a phase transition: they are perfectly interpolable on \(B(0,\rho )\) for \(0<\rho \le 2/\pi \) but exhibit no rigidity for \(\rho >2\).