Hyperuniform random measures, transport and rigidity

TL;DR

This survey reviews the mathematical foundations of hyperuniformity in random measures, covering spectral analysis, examples like determinantal processes, and recent links to optimal transport and rigidity phenomena.

Contribution

It provides a comprehensive framework for hyperuniformity, integrating classical examples with recent advances in transport and rigidity connections.

Findings

Spectral characterizations of hyperuniform measures

Connections between hyperuniformity and optimal transport

Emerging links to rigidity phenomena in random measures

Abstract

This survey explores the foundational theory and recent developments in the study of hyperuniformity. We present a comprehensive mathematical framework in the context of weakly stationary random measures, emphasizing spectral characterizations and second order asymptotics. Classical examples - including determinantal point processes, Gibbs measures, and zero sets of Gaussian analytic functions - are presented in depth to illustrate core principles. We also highlight recent progress connecting hyperuniformity with optimal transport and rigidity phenomena, pointing to emerging directions in the field.