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On strongly rigid hyperfluctuating random measures

Part of: Stochastic processes Geometric probability and stochastic geometry

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: klattm@hhu.de
**Postal address: Karlsruhe Institute of Technology, Institute for Stochastics, 76131 Karlsruhe, Germany. Email address: guenter.last@kit.edu
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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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