Excessive symmetry can preclude cutoff (Q6615424)

An FI-graph is a functor from the category of finite sets with injections to the category of (finite) graphs and graph homomorphisms. Thus an FI-graph may be viewed as a family of nested graphs \(G_{\bullet} = \{G_n\}\), in such a way that each member \(G_n\) is equipped with a natural action by the symmetric group \(S_n\), which is compatible with the inclusions \(G_n \subseteq G_{n+1}\). An FI-graph \(G_{\bullet} = \{G_n\}\) is transitive if for \(n >> 0\), the action of \(S_n\) on \(G_n\) is vertex-transitive. \textit{P. Diaconis} suggested in [Proc. Natl. Acad. Sci. USA 93, No. 4, 1659--1664 (1996; Zbl 0849.60070)] that the cutoff phenomenon seems considerably more likely in situations where the chain has an abundance of symmetry. The authors note that the family of simple random walks on FI-graphs might not exhibit cutoffs, and prove that for a transitive FI-graph \(G_{\bullet} = \{G_n\}\), the family of simple random walks on the graphs \(G_n\) do not exhibit cutoff.