Factor-of-iid Schreier decorations of lattices in Euclidean spaces

A Schreier decoration is a combinatorial coding of an action of the free group Fd 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 Zd,d≥3, the square lattice and also the thre...

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Bibliographic Details
Published inDiscrete mathematics Vol. 347; no. 9; p. 114056
Main Authors Bencs, Ferenc, Hrušková, Aranka, Tóth, László Márton
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.09.2024
Subjects
Archimedean lattice
Factor of iid
Planar lattice
Schreier graph
Site percolation
Transitive graph
Factor of iid
Site percolation
Schreier graph
Transitive graph
Archimedean lattice
Planar lattice
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Summary:A Schreier decoration is a combinatorial coding of an action of the free group Fd 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 Zd,d≥3, the square lattice and also the three other Archimedean lattices of even degree have finitary-factor-of-iid Schreier decorations, and, answering a question of Thornton, 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 the existence of transitive graphs G whose classical chromatic number χ(G) is equal to their factor-of-iid chromatic number.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2024.114056