Random orderings and unique ergodicity of automorphism groups

  • Russell Lyons

    Indiana University, Bloomington, United States
  • Alexander S. Kechris

    California Institute of Technology, Pasadena, United States
  • Omer Angel

    University of British Columbia, Vancouver, Canada


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.

Cite this article

Russell Lyons, Alexander S. Kechris, Omer Angel, Random orderings and unique ergodicity of automorphism groups. J. Eur. Math. Soc. 16 (2014), no. 10, pp. 2059–2095

DOI 10.4171/JEMS/483