Another look at threshold phenomena for random cones

TL;DR

This paper investigates threshold phenomena in high-dimensional stochastic geometry, focusing on the expected face numbers of polyhedral random cones and identifying critical values where behavior changes abruptly.

Contribution

It provides new insights into the critical thresholds and behavior of expected face numbers in high-dimensional random cones, extending previous studies.

Findings

Identification of critical threshold values for face number expectations

Analysis of threshold phenomena for differences in expectations

Extension of threshold phenomena understanding in high-dimensional geometry

Abstract

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.