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Generalized Stealthy Hyperuniform Processes: Maximal Rigidity and the Bounded Holes Conjecture

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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 et al. (Spectra of stationary processes on Z, vol. 261. Birkhäuser, Cham, 2017); in higher dimensions or in the continuum, such a phenomenon seems novel. For stealthy hyperuniform point processes, we prove the Zhang–Stillinger–Torquato conjecture (Zhang et al. in Soft Matter 13:6197–6207, 2017. arXiv:1705.04415) that such processes have bounded holes (empty regions), with a universal bound that depends inversely on the size of U.

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Acknowledgements

The work of J.L.L. was supported in part by the AFOSR Grant FA9550-16-1-0037. The work of S.G. was supported in part by the ARO Grant W911NF-14-1-0094, the NSF Grant DMS-1148711 and the NUS Grant R-146-000-250-133. J.L.L. thanks the Systems Biology group of the Institute for Advanced Studies for their hospitality during this work. The authors thank Salvatore Torquato for valuable discussions, and the anonymous referees for their insightful comments and suggestions.

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Authors and Affiliations

  1. National University of Singapore, Singapore, Singapore

    Subhroshekhar Ghosh

  2. Rutgers University, New Brunswick, USA

    Joel L. Lebowitz

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  1. Subhroshekhar Ghosh
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  2. Joel L. Lebowitz
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Correspondence to Joel L. Lebowitz.

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Communicated by H. Spohn

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Ghosh, S., Lebowitz, J.L. Generalized Stealthy Hyperuniform Processes: Maximal Rigidity and the Bounded Holes Conjecture. Commun. Math. Phys. 363, 97–110 (2018). https://doi.org/10.1007/s00220-018-3226-5

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  • DOI: https://doi.org/10.1007/s00220-018-3226-5

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