Discrete log-concavity and threshold phenomena for atomic measures

TL;DR

This paper explores threshold phenomena for random polytopes generated by atomic measures, providing new justifications, comparing continuous and discrete log-concave settings, and establishing sharp thresholds for lattice p-balls.

Contribution

It offers a missing justification in hypercube threshold arguments, compares continuous and discrete threshold mechanisms, and establishes sharp thresholds for lattice p-balls.

Findings

Identified and justified a key step in hypercube threshold analysis.

Compared continuous and discrete threshold mechanisms.

Established a sharp threshold for lattice p-balls.

Abstract

We investigate threshold phenomena for random polytopes $K_N=\conv\{X_1,\dots,X_N\}$ generated by i.i.d.\ samples from an atomic law $\mu$. We identify and provide a missing justification in the discrete-hypercube threshold argument of Dyer--F\"uredi--McDiarmid, where the supporting half-space estimate is derived via a smooth (gradient/uniqueness) step that can fail at boundary contact points. We then compare threshold-driving mechanisms in the continuous log-concave setting -- through the Cram\'{e}r transform and Tukey's half-space depth -- with their discrete analogues. Within this framework, we establish a sharp threshold for lattice $p$-balls $\mathbb{Z}^n \cap rB_p^n$. Finally, we present structural counterexamples showing that sharp thresholds need not hold in general discrete log-concave settings.