Discrete log-concavity and threshold phenomena for atomic measures
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914272083181568 |
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| author | Brazitikos, Silouanos Pafis, Minas |
| author_facet | Brazitikos, Silouanos Pafis, Minas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15444 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Discrete log-concavity and threshold phenomena for atomic measures Brazitikos, Silouanos Pafis, Minas Probability Metric Geometry 60D05, 60D05, 52A22, 60F10 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. |
| title | Discrete log-concavity and threshold phenomena for atomic measures |
| topic | Probability Metric Geometry 60D05, 60D05, 52A22, 60F10 |
| url | https://arxiv.org/abs/2601.15444 |