Pareto points in growing dimensions
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917352908521472 |
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| author | Ilienko, Andrii Jin, Bochen |
| author_facet | Ilienko, Andrii Jin, Bochen |
| contents | We consider $n$ independent random points uniformly distributed in the $d_n$-dimensional unit cube and study Pareto points, that is, points that do not coordinatewise dominate any other point. We identify the critical growth rate of $d_n$ at which a phase transition occurs: below this threshold, the number of non-Pareto points diverges in probability, whereas above it there are asymptotically no such points. At criticality, the number of non-Pareto points converges in distribution to a Poisson random variable. We further describe their asymptotic spatial distribution in terms of convergence of random point measures.
We also investigate points that dominate exactly $r$ other points and establish analogous phase transitions. For $r=1$, the critical dimension is the same as for non-Pareto points, whereas for every fixed $r\geq 2$ it is different, but, surprisingly, common to all such $r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18698 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pareto points in growing dimensions Ilienko, Andrii Jin, Bochen Probability Primary 60D05, Secondary 60F05, 60G55 We consider $n$ independent random points uniformly distributed in the $d_n$-dimensional unit cube and study Pareto points, that is, points that do not coordinatewise dominate any other point. We identify the critical growth rate of $d_n$ at which a phase transition occurs: below this threshold, the number of non-Pareto points diverges in probability, whereas above it there are asymptotically no such points. At criticality, the number of non-Pareto points converges in distribution to a Poisson random variable. We further describe their asymptotic spatial distribution in terms of convergence of random point measures. We also investigate points that dominate exactly $r$ other points and establish analogous phase transitions. For $r=1$, the critical dimension is the same as for non-Pareto points, whereas for every fixed $r\geq 2$ it is different, but, surprisingly, common to all such $r$. |
| title | Pareto points in growing dimensions |
| topic | Probability Primary 60D05, Secondary 60F05, 60G55 |
| url | https://arxiv.org/abs/2603.18698 |