Daniel Hug Karlsruhe Institute of Technology (KIT), Department of Mathematics, D-76128 Karlsruhe, Germany

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Rolf Schneider Mathematisches Institut, Albert-Ludwigs-Universität, D-79104 Freiburg i. Br., Germany

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In stochastic geometry there are several instances of threshold phenomena in high dimensions: the behavior of a limit of some expectation changes abruptly when some parameter passes through a critical value. This note continues the investigation of the expected face numbers of polyhedral random cones, when the dimension of the ambient space increases to infinity. In the focus are the critical values of the observed threshold phenomena, as well as threshold phenomena for differences instead of quotients.

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    D. Amelunxen, M. Lotz, M. B. McCoy and J. A. Tropp. Living on the edge: phase transitions in convex programs with random data. Inf. Inference, 3:224294, 2014.

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    T. M. Cover and B. Efron. Geometrical probability and random points on a hypersphere. Ann. Math. Stat., 38:213220, 1967.

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    O. Devillers, M. Glisse, X. Goaoc, G. Moroz and M. Reitzner. The monotonicity of f-vectors of random polytopes. Electron. Commun. Probab., 18(23):8 pp, 2013.

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    D. L. Donoho and J. Tanner. Counting the faces of randomly-projected hypercubes and orthants, with applications. Discrete Comput. Geom., 43:522541, 2010.

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    T. Godland, Z. Kabluchko and C. Thäle. Random cones in high dimensions I: Donoho– Tanner and Cover–Efron cones. arXiv:2012.06189v1

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    D. Hug and R. Schneider. Random conical tessellations. Discrete Comput. Geom., 56:395426, 2016.

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    D. Hug and R. Schneider. Threshold phenomena for random cones. Discrete Comput. Geom., accepted. arXiv:2004.11473v2.

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    M. Okamoto. Some inequalities relating to the partial sum of binomial probabilities. Ann. Inst. Statist. Math., 10:2935, 1958.

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    R. Schneider and W. Weil. Stochastic and Integral Geometry. Springer, Berlin, 2008.

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    J. G. Wendel. A problem in geometric probability. Math. Scand., 11:109111, 1962.

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