Factor-of-iid Schreier decorations of lattices in Euclidean spaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866912586522427392 |
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| author | Bencs, Ferenc Hrušková, Aranka Tóth, László Márton |
| author_facet | Bencs, Ferenc Hrušková, Aranka Tóth, László Márton |
| contents | A Schreier decoration is a combinatorial coding of an action of the free group $F_d$ on the vertex set of a $2d$-regular graph. We investigate whether a Schreier decoration exists on various countably infinite transitive graphs as a factor of iid.
We show that $\mathbb{Z}^d,d\geq3$, the square lattice and also the three other Archimedean lattices of even degree have finitary-factor-of-iid Schreier decorations, and exhibit examples of transitive graphs of arbitrary even degree in which obtaining such a decoration as a factor of iid is impossible.
We also prove that symmetrical planar lattices with all degrees even have a factor of iid balanced orientation, meaning the indegree of every vertex is equal to its outdegree, and demonstrate that the property of having a factor-of-iid balanced orientation is not invariant under quasi-isometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_12577 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Factor-of-iid Schreier decorations of lattices in Euclidean spaces Bencs, Ferenc Hrušková, Aranka Tóth, László Márton Probability Combinatorics Dynamical Systems 05E18, 05C63, 37A30, 60C05 A Schreier decoration is a combinatorial coding of an action of the free group $F_d$ on the vertex set of a $2d$-regular graph. We investigate whether a Schreier decoration exists on various countably infinite transitive graphs as a factor of iid. We show that $\mathbb{Z}^d,d\geq3$, the square lattice and also the three other Archimedean lattices of even degree have finitary-factor-of-iid Schreier decorations, and exhibit examples of transitive graphs of arbitrary even degree in which obtaining such a decoration as a factor of iid is impossible. We also prove that symmetrical planar lattices with all degrees even have a factor of iid balanced orientation, meaning the indegree of every vertex is equal to its outdegree, and demonstrate that the property of having a factor-of-iid balanced orientation is not invariant under quasi-isometry. |
| title | Factor-of-iid Schreier decorations of lattices in Euclidean spaces |
| topic | Probability Combinatorics Dynamical Systems 05E18, 05C63, 37A30, 60C05 |
| url | https://arxiv.org/abs/2101.12577 |