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Published 2014 | Submitted
Journal Article Open

Random orderings and unique ergodicity of automorphism groups

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 and Weiss'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.

Additional Information

© 2014 European Mathematical Society. Received August 12, 2012 and in revised form November 19, 2012. We would like to thank Gregory Cherlin and the anonymous referee for many useful comments. Research of O. Angel partially supported by NSERC and the Sloan Foundation. Research of A. S. Kechris partially supported by NSF grant DMS-0968710. Research of R. Lyons partially supported by NSF grant DMS-1007244 and Microsoft Research.

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August 19, 2023
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