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Discrete log-concavity and threshold phenomena for atomic measures

Silouanos Brazitikos, Minas Pafis

TL;DR

This work analyzes threshold phenomena for random polytopes generated from atomic measures, linking geometric hulls to Tukey half-space depth and large-deviation rate functions. It first identifies and fixes a gap in the Dyer–Füredi–McDiarmid approach for the discrete cube, then develops a positive theory under independence for symmetric, endpoint-mass marginals, and finally exposes a discrete-convex framework with counterexamples showing sharp thresholds can fail without structural assumptions. The paper proves a weak threshold for broad classes of product measures and a sharp threshold in lattice settings, including a complete analysis of the discrete cube and a polynomial-scale threshold for lattice points in ell_p-balls. It also constructs counterexamples demonstrating the necessity of independence and integrability assumptions and shows that discrete log-concave settings can behave very differently from the continuous case. Overall, the results delineate when threshold phenomena persist in discrete settings and when they fail, clarifying the roles of independence, endpoint mass, and integrability in controlling the hull-mass dynamics of random polytopes.

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 $μ$. We identify and provide a missing justification in the discrete-hypercube threshold argument of Dyer--Füredi--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 Cramér 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.

Discrete log-concavity and threshold phenomena for atomic measures

TL;DR

This work analyzes threshold phenomena for random polytopes generated from atomic measures, linking geometric hulls to Tukey half-space depth and large-deviation rate functions. It first identifies and fixes a gap in the Dyer–Füredi–McDiarmid approach for the discrete cube, then develops a positive theory under independence for symmetric, endpoint-mass marginals, and finally exposes a discrete-convex framework with counterexamples showing sharp thresholds can fail without structural assumptions. The paper proves a weak threshold for broad classes of product measures and a sharp threshold in lattice settings, including a complete analysis of the discrete cube and a polynomial-scale threshold for lattice points in ell_p-balls. It also constructs counterexamples demonstrating the necessity of independence and integrability assumptions and shows that discrete log-concave settings can behave very differently from the continuous case. Overall, the results delineate when threshold phenomena persist in discrete settings and when they fail, clarifying the roles of independence, endpoint mass, and integrability in controlling the hull-mass dynamics of random polytopes.

Abstract

We investigate threshold phenomena for random polytopes generated by i.i.d.\ samples from an atomic law . We identify and provide a missing justification in the discrete-hypercube threshold argument of Dyer--Füredi--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 Cramér transform and Tukey's half-space depth -- with their discrete analogues. Within this framework, we establish a sharp threshold for lattice -balls . Finally, we present structural counterexamples showing that sharp thresholds need not hold in general discrete log-concave settings.
Paper Structure (17 sections, 46 theorems, 206 equations)

This paper contains 17 sections, 46 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Thresholds and the half-space mechanism.
  3. The rate-function viewpoint.
  4. The issue in the discrete cube proof of DFM.
  5. Two directions in this paper (one positive, one negative).
  6. Examples and calibration.
  7. Notation and main statements
  8. Discrete log-concave random vectors
  9. Weak threshold for product measures
  10. Threshold for product measure with mass at the endpoint
  11. Discrete log-concavity and threshold
  12. Sharp threshold in lattice settings
  13. Discrete hypercube $\{0,1\}^n$: complete analysis
  14. Lattice points in $\ell_p$-ball
  15. Counterexamples
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $X$ be a discrete log-concave random vector in $\mathbb{Z}^n$ with p.m.f $p:\mathbb{Z}^n \to [0,1)$ and let $f$ be its log-concave extension. If $f$ is integrable, then for every $q \geq 1$ we have

Theorems & Definitions (93)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 3.1: Discrete log-concavity on $\mathbb{Z}$
  • Definition 3.2: convex-extensible
  • Lemma 3.3
  • proof
  • Definition 3.4: Discrete log-concave random vectors
  • ...and 83 more