[Paper Review] Random Orderings and Unique Ergodicity of Automorphism Groups

Omer Angel, Alexander S. Kechris+1 more|arXiv (Cornell University)|Aug 12, 2012

Advanced Topology and Set Theory8 references4 citations

TL;DR

This paper establishes that the only consistent random ordering for finite graphs, $K_n$-free graphs, $r$-uniform hypergraphs, and metric spaces with distances in a given additive subsemigroup of $\mathbb{R}^+$ is the uniform random ordering. This result implies the unique ergodicity of the automorphism group of the random graph and provides the first non-compact, non-extremely amenable examples of uniquely ergodic groups beyond Glasner and Weiss’s example.

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.

Motivation & Objective

• To determine whether consistent random orderings on finite structures—such as graphs, hypergraphs, and metric spaces—must necessarily be uniform.
• To investigate the implications of such random orderings for the unique ergodicity of automorphism groups.
• To identify new classes of Polish groups that are uniquely ergodic, beyond compact or extremely amenable groups.
• To explore the existence of non-uniform consistent random orderings in restricted classes of graphs and metric spaces, such as bounded-degree graphs or Euclidean metric spaces.
• To establish quantitative bounds on the total variation distance between non-uniform and uniform orderings in finite structures.

Proposed method

• Define a consistent random ordering as a family of probability measures $\mu_G$ on linear orderings of vertex sets of finite structures $G$, satisfying isomorphism invariance and restriction consistency.
• Use model-theoretic and combinatorial techniques, including the Ramsey property and the expansion property, to analyze the structure of such orderings.
• Apply the theory of Fraïssé limits and universal minimal flows to link consistent random orderings to invariant measures on dynamical systems.
• Employ the random projection method (suggested by Leonard Schulman) to construct non-uniform consistent orderings in classes like bounded-degree graphs and Euclidean metric spaces.
• Derive quantitative total variation bounds using probabilistic and extremal combinatorial arguments, particularly for $r$-uniform hypergraphs.
• Use the uniformity of the measure $\mu(N_{{\boldsymbol{A}},<}) = \frac{1}{k({\boldsymbol{A}})}$ on the space of admissible orderings to analyze uniqueness of invariant measures.

Experimental results

Research questions

• RQ1 Is the uniform random ordering the only consistent random ordering for the class of all finite graphs?
• RQ2 Can non-uniform consistent random orderings exist for certain restricted classes of graphs or metric spaces, such as bounded-degree graphs or Euclidean metric spaces?
• RQ3 Does the existence of a consistent random ordering imply unique ergodicity of the automorphism group of the Fraïssé limit?
• RQ4 What is the maximal total variation distance between a non-uniform and the uniform random ordering on finite structures of size at most $n$?
• RQ5 Is the unique invariant measure on the universal minimal flow of an amenable automorphism group of a Fraïssé structure always uniform over admissible orderings?

Key findings

• The only consistent random ordering for the class of all finite graphs, $K_n$-free graphs, $r$-uniform hypergraphs, and metric spaces with distances in a fixed additive subsemigroup of $\mathbb{R}^+$ is the uniform random ordering.
• This result implies the unique ergodicity of the automorphism group of the random graph, providing the first non-compact, non-extremely amenable example of a uniquely ergodic Polish group beyond Glasner and Weiss’s example.
• For $r$-uniform hypergraphs of size at most $n$, the total variation distance between any consistent random ordering and the uniform one is bounded by $C\sqrt{\frac{\log n}{n^{r-1}}}$ for some constant $C = C(k,r)$.
• In the case of graphs, a construction yields a consistent random ordering with total variation distance at least $C/n$ from uniform on some subgraphs.
• Non-uniform consistent random orderings exist for classes such as bounded-degree graphs and Euclidean metric spaces, constructed via a random projection method.
• The unique invariant measure on the universal minimal flow of an amenable automorphism group of a Fraïssé structure is conjectured to be uniform over admissible orderings, and this holds in all known cases.