Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912722894979072 |
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| author | Basdevant, Anne-Laure Gerin, Lucas Marivain, Maxime |
| author_facet | Basdevant, Anne-Laure Gerin, Lucas Marivain, Maxime |
| contents | A famous result by Hammersley and Versik-Kerov states that the length $L_n$ of the longest increasing subsequence among $n$ iid continuous random variables grows like $2\sqrt{n}$. We investigate here the asymptotic behavior of $L_n$ for distributions with atoms. For purely discrete random variables, we characterize the asymptotic order of $L_n$ through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between $\mathcal{O}(1)$ and $o(\sqrt{n})$, depending on the tail behavior of the distribution. We can then easily deduce the asymptotics of $L_n$ for a completely arbitrary distribution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05775 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process Basdevant, Anne-Laure Gerin, Lucas Marivain, Maxime Combinatorics Probability A famous result by Hammersley and Versik-Kerov states that the length $L_n$ of the longest increasing subsequence among $n$ iid continuous random variables grows like $2\sqrt{n}$. We investigate here the asymptotic behavior of $L_n$ for distributions with atoms. For purely discrete random variables, we characterize the asymptotic order of $L_n$ through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between $\mathcal{O}(1)$ and $o(\sqrt{n})$, depending on the tail behavior of the distribution. We can then easily deduce the asymptotics of $L_n$ for a completely arbitrary distribution. |
| title | Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2507.05775 |