Generalized stealthy hyperuniform processes : maximal rigidity and the bounded holes conjecture

Abstract

We study translation invariant stochastic processes on ${\mathbb{R}}^d$ or ${\mathbb{Z}}^d$ whose diffraction spectrum or structure function $S(k)$, i.e. the Fourier transform of the truncated total pair correlation function, vanishes on an open set $U$ in the wave space. A key family of such processes are stealthy hyperuniform point processes, for which the origin $k=0$ is in $U$; these are of much current physical interest. We show that all such processes exhibit the following remarkable maximal rigidity : namely, the configuration outside a bounded region determines, with probability 1, the exact value (or the exact locations of the points) of the process inside the region. In particular, such processes are completely determined by their tail. In the 1D discrete setting (i.e. $\mathbb{Z}$-valued processes on $\mathbb{Z}$), this can also be seen as a consequence of a recent theorem of Borichev, Sodin and Weiss; in higher dimensions or in the continuum, such a phenomenon seems novel. For stealthy hyperuniform point processes, we prove the Zhang-Stillinger-Torquato conjecture that such processes have bounded holes (empty regions), with a universal bound that depends inversely on the size of $U$.

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• Effective Elastic Wave Characteristics of Composite Media
• Theoretical Prediction of the Effective Dynamic Dielectric Constant of Disordered Hyperuniform Anisotropic Composites Beyond the Long-Wavelength Regime
• Generating large disordered stealthy hyperuniform systems with ultra-high accuracy to determine their physical properties
• Dynamic Measure of Hyperuniformity and Nonhyperuniformity in Heterogeneous Media via the Diffusion Spreadability
• Realizable hyperuniform and nonhyperuniform particle configurations with targeted spectral functions via effective pair interactions
• Light transport through amorphous photonic materials with localization and bandgap regimes
• Characterizing the Hyperuniformity of Ordered and Disordered Two-Phase Media
• Rigidity hierarchy in random point fields: random polynomials and determinantal processes
• Scaling of diffraction intensities near the origin: Some rigorous results
• Low-temperature statistical mechanics of the QuanTizer problem: fast quenching and equilibrium cooling of the three-dimensional Voronoi Liquid
• Fluctuation and Entropy in Spectrally Constrained random fields
• Hyperuniformity Classes of Quasiperiodic Tilings via Diffusion Spreadability
• Dynamical properties of particulate composites derived from ultradense stealthy hyperuniform sphere packings
• Existence of Nonequilibrium Glasses in the Degenerate Stealthy Hyperuniform Ground-State Manifold
• Entanglement entropy and hyperuniformity of Ginibre and Weyl-Heisenberg ensembles
• Nearest-Neighbor Functions for Disordered Stealthy Hyperuniform Many-Particle Systems
• Impact of Random Spatial Truncation and Reciprocal-Space Binning on the Detection of Hyperuniformity in Disordered Systems
• Spectral rigidity of random Schrödinger operators via Feynman-Kac formulas
• On Spatial Conditioning of the Spectrum of Discrete Random Schrödinger Operators
• Rigidity for zero sets of Gaussian entire functions
• Precise Determination of the Long-Time Asymptotics of the Diffusion Spreadability of Two-Phase Media