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Hyperuniform random measures, transport and rigidity

Raphaël Lachièze-Rey

TL;DR

This survey builds a comprehensive framework for hyperuniformity across weakly stationary random measures and point processes, linking second-order statistics to spectral structure and transport phenomena. It highlights three emblematic, mathematically tractable models—Sine_beta, Ginibre, and planar GAF zeros—and explicates their hyperuniformity, rigidity, and transport properties, including optimal transport and allocation perspectives. The work then develops a transport/perturbation bridge showing how hyperuniform processes can be viewed as perturbed lattices and, conversely, how perturbed lattices give rise to hyperuniformity, with a deep dive into rigidity, stealthy regimes, and quasicrystal models. It concludes with quantitative tools, estimation procedures, and simulation strategies (structure-factor package, HU tests) to detect and capitalize on hyperuniformity in finite samples and applications, underscoring the practical impact in numerical integration, texture synthesis, and disordered materials. The results unify spectral, probabilistic, and transport viewpoints, establishing a robust foundation for future investigations into hyperuniform transport, rigidity, and their applications.

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.

Hyperuniform random measures, transport and rigidity

TL;DR

This survey builds a comprehensive framework for hyperuniformity across weakly stationary random measures and point processes, linking second-order statistics to spectral structure and transport phenomena. It highlights three emblematic, mathematically tractable models—Sine_beta, Ginibre, and planar GAF zeros—and explicates their hyperuniformity, rigidity, and transport properties, including optimal transport and allocation perspectives. The work then develops a transport/perturbation bridge showing how hyperuniform processes can be viewed as perturbed lattices and, conversely, how perturbed lattices give rise to hyperuniformity, with a deep dive into rigidity, stealthy regimes, and quasicrystal models. It concludes with quantitative tools, estimation procedures, and simulation strategies (structure-factor package, HU tests) to detect and capitalize on hyperuniformity in finite samples and applications, underscoring the practical impact in numerical integration, texture synthesis, and disordered materials. The results unify spectral, probabilistic, and transport viewpoints, establishing a robust foundation for future investigations into hyperuniform transport, rigidity, and their applications.

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.
Paper Structure (78 sections, 58 theorems, 228 equations, 4 figures)

This paper contains 78 sections, 58 theorems, 228 equations, 4 figures.

Table of Contents

  1. Introduction and examples
  2. Point processes formalism and first definition of HU
  3. Perturbed lattices
  4. Disordered samples
  5. Three emblematic examples
  6. The $\text{\rm{Sine}}_{ \beta }$ processes.
  7. The Ginibre ensemble
  8. Zeros of the planar GAF
  9. Global order and local disorder
  10. Mathematical hyperuniformity of random measures
  11. Wide sense stationary random measures
  12. Spectral characterisation of hyperuniformity
  13. Universal variance lower bounds and non-spherical windows
  14. Hyperuniformity exponent and classification
  15. CLTs and Brillinger mixing
  16. ...and 63 more sections

Key Result

Theorem 1.1

For $\beta >0,$ there is a stationary point process $\mathsf{P}^{ \beta }\subset \mathbb{R}$, called $\text{\rm{Sine}}_{ \beta }$ process, such that Furthermore, $\mathsf{P}^{ \beta }$ is hyperuniform.

Figures (4)

  • Figure 1: Left. Disordered hyperuniform receptors avian-eyes. Middle. Periodic "ordered" photoreceptors. Right. Dithering - Greyscale levels replaced by hyperuniform "blue noise" samples deGoes, ACM Trans. Graph.
  • Figure 2: Left. Ginibre ensemble. Right. GAF zeros.
  • Figure 3: Sending kids to school. Left. Ginibre eigenvalues. Middle. Poisson points. Right. GAF zeros. Ilustration by D. Hawat
  • Figure :

Theorems & Definitions (107)

  • Definition 1.1
  • Example 1.1: IPL
  • Definition 1.2
  • Example 1.2
  • Theorem 1.1: ValkoVirag
  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • ...and 97 more