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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2412.19730 |
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| _version_ | 1866912212508999680 |
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| author | Borga, Jacopo Lin, Andrew |
| author_facet | Borga, Jacopo Lin, Andrew |
| contents | Permutons, which are probability measures on the unit square $[0, 1]^2$ with uniform marginals, are the natural scaling limits for sequences of (random) permutations.
We introduce a $d$-dimensional generalization of these measures for all $d \ge 2$, which we call $d$-dimensional permutons, and extend -- from the two-dimensional setting -- the theory to prove convergence of sequences of (random) $d$-dimensional permutations to (random) $d$-dimensional permutons.
Building on this new theory, we determine the random high-dimensional permuton limits for two natural families of high-dimensional permutations. First, we determine the $3$-dimensional permuton limit for Schnyder wood permutations, which bijectively encode planar triangulations decorated by triples of spanning trees known as Schnyder woods. Second, we identify the $d$-dimensional permuton limit for $d$-separable permutations, a pattern-avoiding class of $d$-dimensional permutations generalizing ordinary separable permutations.
Both high-dimensional permuton limits are random and connected to previously studied universal 2-dimensional permutons, such as the Brownian separable permutons and the skew Brownian permutons, and share interesting connections with objects arising from random geometry, including the continuum random tree, Schramm--Loewner evolutions, and Liouville quantum gravity surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19730 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | High-dimensional permutons: theory and applications Borga, Jacopo Lin, Andrew Probability Mathematical Physics Combinatorics Permutons, which are probability measures on the unit square $[0, 1]^2$ with uniform marginals, are the natural scaling limits for sequences of (random) permutations. We introduce a $d$-dimensional generalization of these measures for all $d \ge 2$, which we call $d$-dimensional permutons, and extend -- from the two-dimensional setting -- the theory to prove convergence of sequences of (random) $d$-dimensional permutations to (random) $d$-dimensional permutons. Building on this new theory, we determine the random high-dimensional permuton limits for two natural families of high-dimensional permutations. First, we determine the $3$-dimensional permuton limit for Schnyder wood permutations, which bijectively encode planar triangulations decorated by triples of spanning trees known as Schnyder woods. Second, we identify the $d$-dimensional permuton limit for $d$-separable permutations, a pattern-avoiding class of $d$-dimensional permutations generalizing ordinary separable permutations. Both high-dimensional permuton limits are random and connected to previously studied universal 2-dimensional permutons, such as the Brownian separable permutons and the skew Brownian permutons, and share interesting connections with objects arising from random geometry, including the continuum random tree, Schramm--Loewner evolutions, and Liouville quantum gravity surfaces. |
| title | High-dimensional permutons: theory and applications |
| topic | Probability Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2412.19730 |