A NONAMENABLE “FACTOR” OF A EUCLIDEAN SPACE

Ádám Timár

doi:10.1214/20-AOP1485

A NONAMENABLE “FACTOR” OF A EUCLIDEAN SPACE

Ádám Timár

Science Institute

University of Iceland

Research output: Contribution to journal › Article › peer-review

Abstract

Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space R^d, d ≥ 3, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in R d as an isometry-invariant random partition of R^d to bounded polyhedra, and also as an isometry-invariant random partition of R d to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of IID’s.

Original language	English
Pages (from-to)	1427-1449
Number of pages	23
Journal	Annals of Probability
Volume	49
Issue number	3
DOIs	https://doi.org/10.1214/20-AOP1485
Publication status	Published - May 2021

Bibliographical note

Funding Information: This work was started at the Bernoulli Center (CIB) conference “Statistical physics on transitive graphs.” I would like to thank Itai Benjamini, Dorottya Beringer, Damien Gaboriau, Russ Lyons, Gábor Pete, Mikael De La Salle and Romain Tessera for inspiring conversations, and a referee for useful comments. The author was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme, by the Hungarian National Research, Development and Innovation Office, NKFIH Grant K109684, and by Grant LP 2016-5 of the Hungarian Academy of Sciences. After finishing but before publishing this paper, the author started working part-time at the University of Iceland. Funding Information: The author was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme, by the Hungarian National Research, Development and Innovation Office, NKFIH Grant K109684, and by Grant LP 2016-5 of the Hungarian Academy of Sciences. After finishing but before publishing this paper, the author started working part-time at the University of Iceland. Publisher Copyright: © Institute of Mathematical Statistics, 2021

Other keywords

Random tiling
factor of IID
indistinguishability
isometry-invariant tiling

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Cite this

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title = "A NONAMENABLE “FACTOR” OF A EUCLIDEAN SPACE",

abstract = "Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space Rd, d ≥ 3, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in R d as an isometry-invariant random partition of Rd to bounded polyhedra, and also as an isometry-invariant random partition of R d to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of IID{\textquoteright}s.",

keywords = "Random tiling, factor of IID, indistinguishability, isometry-invariant tiling",

author = "{\'A}d{\'a}m Tim{\'a}r",

note = "Funding Information: This work was started at the Bernoulli Center (CIB) conference “Statistical physics on transitive graphs.” I would like to thank Itai Benjamini, Dorottya Beringer, Damien Gaboriau, Russ Lyons, G{\'a}bor Pete, Mikael De La Salle and Romain Tessera for inspiring conversations, and a referee for useful comments. The author was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme, by the Hungarian National Research, Development and Innovation Office, NKFIH Grant K109684, and by Grant LP 2016-5 of the Hungarian Academy of Sciences. After finishing but before publishing this paper, the author started working part-time at the University of Iceland. Funding Information: The author was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme, by the Hungarian National Research, Development and Innovation Office, NKFIH Grant K109684, and by Grant LP 2016-5 of the Hungarian Academy of Sciences. After finishing but before publishing this paper, the author started working part-time at the University of Iceland. Publisher Copyright: {\textcopyright} Institute of Mathematical Statistics, 2021",

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KW - indistinguishability

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JO - Annals of Probability

JF - Annals of Probability

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ER -

A NONAMENABLE “FACTOR” OF A EUCLIDEAN SPACE

Abstract

Bibliographical note

Other keywords

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