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On strongly rigid hyperfluctuating random measures

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  15 August 2022

Michael Andreas Klatt  and
Günter Last
Michael Andreas Klatt*
Affiliation:
Heinrich-Heine-University Düsseldorf
Günter Last*
Affiliation:
Karlsruhe Institute of Technology
*
*Postal address: Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany; Experimental Physics, Saarland University, Center for Biophysics, 66123 Saarbrücken, Germany. Email address: [email protected]
**Postal address: Karlsruhe Institute of Technology, Institute for Stochastics, 76131 Karlsruhe, Germany. Email address: [email protected]
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Abstract

In contrast to previous belief, we provide examples of stationary ergodic random measures that are both hyperfluctuating and strongly rigid. Therefore we study hyperplane intersection processes (HIPs) that are formed by the vertices of Poisson hyperplane tessellations. These HIPs are known to be hyperfluctuating, that is, the variance of the number of points in a bounded observation window grows faster than the size of the window. Here we show that the HIPs exhibit a particularly strong rigidity property. For any bounded Borel set B, an exponentially small (bounded) stopping set suffices to reconstruct the position of all points in B and, in fact, all hyperplanes intersecting B. Therefore the random measures supported by the hyperplane intersections of arbitrary (but fixed) dimension, are also hyperfluctuating. Our examples aid the search for relations between correlations, density fluctuations, and rigidity properties.

Keywords

Poisson hyperplane tessellationshyperplane intersection processesstrong rigidityhyperfluctuationhyperuniformity

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 60G57: Random measures
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 948 - 961
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Baumstark, V. and Last, G. (2009). Gamma distributions for stationary Poisson flat processes. Adv. Appl. Prob. 41, 911–939.CrossRefGoogle Scholar
Bufetov, A. I. (2016). Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel. Bull. Math. Sci. 6.1, 163–172.CrossRefGoogle Scholar
Dereudre, D., Hardy, A., Leblé, T. and Maïda, M. (2020). DLR equations and rigidity for the Sine-beta process. Commun. Pure Appl. Math. 74, 172–222.CrossRefGoogle Scholar
Ghosh, S. and Krishnapur, M. (2021). Rigidity hierarchy in random point fields: random polynomials and determinantal processes. Commun. Math. Phys. 388, 1205–1234.CrossRefGoogle Scholar
Ghosh, S. and Lebowitz, J. L. (2017). Fluctuations, large deviations and rigidity in hyperuniform systems: a brief survey. Indian J. Pure Appl. Math. 48, 609–631.CrossRefGoogle Scholar
Ghosh, S. and Lebowitz, J. L. (2017). Number rigidity in superhomogeneous random point fields. J. Statist. Phys. 166, 1016–1027.CrossRefGoogle Scholar
Ghosh, S. and Lebowitz, J. L. (2018). Generalized stealthy hyperuniform processes: maximal rigidity and the bounded holes conjecture. Commun. Math. Phys. 363, 97–110.CrossRefGoogle Scholar
Ghosh, S. and Peres, Y. (2017). Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues. Duke Math. J. 166, 1789–1858.CrossRefGoogle Scholar
Heinrich, L., Schmidt, H. and Schmidt, V. (2006). Central limit theorems for Poisson hyperplane tessellations. Ann. Appl. Prob. 16, 919–950.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Klatt, M. A., Last, G. and Yogeshwaran, D. (2020). Hyperuniform and rigid stable matchings. Random Struct. Algorithms 57, 439–473.CrossRefGoogle Scholar
Lachièze-Rey, R. (2020). Variance linearity for real Gaussian zeros. Ann. Inst. H. Poincaré Prob. Statist., to appear. Available at arXiv:2006.10341.Google Scholar
Last, G. and Penrose, M. (2017). Lectures on the Poisson Process. Cambridge University Press.CrossRefGoogle Scholar
Last, G., Penrose, M. P., Schulte, M. and Thäle, C. (2014). Moments and central limit theorems for some multivariate Poisson functionals. Adv. Appl. Prob. 46, 348–364.CrossRefGoogle Scholar
Molchanov, I. (2017). Theory of Random Sets, 2nd edn. Springer, London.CrossRefGoogle Scholar
Peres, Y. and Sly, A. (2014). Rigidity and tolerance for perturbed lattices. Available at arXiv:1409.4490.Google Scholar
Schneider, R. (2019). Interaction of Poisson hyperplane processes and convex bodies. J. Appl. Prob. 56, 1020–1032.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar
Torquato, S. (2018). Hyperuniform states of matter. Phys. Rep. 745, 1–95.CrossRefGoogle Scholar
Torquato, S. and Stillinger, F. H. (2003). Local density fluctuations, hyperuniformity, and order metrics. Phys. Rev. E 68, 41–113.Google ScholarPubMed