# Random orderings and unique ergodicity of automorphism groups

Omer Angel; Alexander S. Kechris; Russell Lyons

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 10, page 2059-2095
- ISSN: 1435-9855

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topAngel, Omer, Kechris, Alexander S., and Lyons, Russell. "Random orderings and unique ergodicity of automorphism groups." Journal of the European Mathematical Society 016.10 (2014): 2059-2095. <http://eudml.org/doc/277464>.

@article{Angel2014,

abstract = {We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner andWeiss’s example of the group of all permutations of the integers. We also contrast these results to those for certain special classes of graphs and metric spaces in which such random orderings can be found that are not uniform.},

author = {Angel, Omer, Kechris, Alexander S., Lyons, Russell},

journal = {Journal of the European Mathematical Society},

keywords = {graphs; hypergraphs; the random graph; metric spaces; Fraïssé; Ramsey; minimal flow; Urysohn space; graphs; random graph; metric spaces; minimal flow},

language = {eng},

number = {10},

pages = {2059-2095},

publisher = {European Mathematical Society Publishing House},

title = {Random orderings and unique ergodicity of automorphism groups},

url = {http://eudml.org/doc/277464},

volume = {016},

year = {2014},

}

TY - JOUR

AU - Angel, Omer

AU - Kechris, Alexander S.

AU - Lyons, Russell

TI - Random orderings and unique ergodicity of automorphism groups

JO - Journal of the European Mathematical Society

PY - 2014

PB - European Mathematical Society Publishing House

VL - 016

IS - 10

SP - 2059

EP - 2095

AB - We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner andWeiss’s example of the group of all permutations of the integers. We also contrast these results to those for certain special classes of graphs and metric spaces in which such random orderings can be found that are not uniform.

LA - eng

KW - graphs; hypergraphs; the random graph; metric spaces; Fraïssé; Ramsey; minimal flow; Urysohn space; graphs; random graph; metric spaces; minimal flow

UR - http://eudml.org/doc/277464

ER -

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