The global and local limit of the continuous-time Mallows process
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929312436846592 |
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| author | Adamczak, Radosław Kotowski, Michał |
| author_facet | Adamczak, Radosław Kotowski, Michał |
| contents | Continuous-time Mallows processes are processes of random permutations of the set $\{1, \ldots, n\}$ whose marginal at time $t$ is the Mallows distribution with parameter $t$. Recently Corsini showed that there exists a unique Markov Mallows process whose left inversions are independent counting processes. We prove that this process admits a global and a local limit as $n \to \infty$. The global limit, obtained after suitably rescaling space and time, is an explicit stochastic process on $[0,1]$ whose description is based on the permuton limit of the Mallows distribution, analyzed by Starr. The local limit is a process of permutations of $\mathbb{Z}$ which is closely related to the construction of the Mallows distribution on permutations of $\mathbb{Z}$ due to Gnedin and Olshanski. Our results demonstrate an analogy between the asymptotic behavior of Mallows processes and the recently studied limiting properties of random sorting networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08554 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The global and local limit of the continuous-time Mallows process Adamczak, Radosław Kotowski, Michał Probability Combinatorics 60C05, 60J27, 60J75, 05A05 Continuous-time Mallows processes are processes of random permutations of the set $\{1, \ldots, n\}$ whose marginal at time $t$ is the Mallows distribution with parameter $t$. Recently Corsini showed that there exists a unique Markov Mallows process whose left inversions are independent counting processes. We prove that this process admits a global and a local limit as $n \to \infty$. The global limit, obtained after suitably rescaling space and time, is an explicit stochastic process on $[0,1]$ whose description is based on the permuton limit of the Mallows distribution, analyzed by Starr. The local limit is a process of permutations of $\mathbb{Z}$ which is closely related to the construction of the Mallows distribution on permutations of $\mathbb{Z}$ due to Gnedin and Olshanski. Our results demonstrate an analogy between the asymptotic behavior of Mallows processes and the recently studied limiting properties of random sorting networks. |
| title | The global and local limit of the continuous-time Mallows process |
| topic | Probability Combinatorics 60C05, 60J27, 60J75, 05A05 |
| url | https://arxiv.org/abs/2404.08554 |