Maximal rigidity of random measure and uniqueness pairs: stealthy processes, quasicrystals and periodicity

TL;DR

This paper explores maximal rigidity in spatial processes, demonstrating how certain spectral properties lead to perfect interpolation and periodicity, unifying results across quasicrystals, stealthy processes, and integer-valued models.

Contribution

It extends classical one-dimensional spectral rigidity results to higher dimensions and continuous settings, connecting them with the uncertainty principle and identifying phase transitions in process rigidity.

Findings

Maximal rigidity occurs for processes with spectral gaps or deep zeros.

Discrete integer-valued processes are necessarily periodic with certain spectral conditions.

A phase transition in rigidity behavior is identified at a critical radius.

Abstract

This article investigates the phenomenon of maximal rigidity in spatial processes, 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 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: quasicrystals and stealthy processes are shown to be maximally rigid on cones, and 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 $\rho$ ___ 2 $\pi$ but exhibit no rigidity for $\rho$ > 2.