Inferring traits of hyperuniformity from local structures via persistent homology

Abstract: Hyperuniformity refers to the suppression of density fluctuations at large scales. Typical for ordered systems, this property also emerges in several disordered physical and biological systems, where it is particularly relevant to understand mechanisms of pattern formation and to exploit peculiar attributes, e.g., interaction with light and transport phenomena. While hyperuniformity is a global property, ideally defined for infinitely extended systems, several disordered correlated systems have finite size. It has been shown in [Phys. Rev. Research 6, 023107 (2024)] that global hyperuniform characteristics systematically correlate with distributions of topological properties representative of local arrangements. In this work, building on this information, we explore and assess the inverse relationship between hyperuniformity and local structures in point patterns as described by persistent homology. Standard machine learning algorithms trained on persistence diagrams are shown to detect hyperuniformity of periodic point patterns with high accuracy. Therefore, we demonstrate that the information on patterns' local structures allows for characterizing whether finite size arrangements are analogous to those realized in hyperuniform patterns. Then, addressing more quantitative aspects, we show that parameters defining hyperuniformity globally can be reconstructed by comparing persistence diagrams of targeted patterns with reference ones. We also explore the generation of patterns entailing given topological properties. The results of this study pave the way for advanced analysis of hyperuniform patterns including local information, and introduce basic concepts for their inverse design.