Random orderings and unique ergodicity of automorphism groups

Angel, 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 -