Excessive symmetry can preclude cutoff

In this paper we look at the families of random walks arising from FI-graphs. One may think of these objects as families of nested graphs, each equipped with a natural action by a symmetric group S_n, such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in [9], while a systematic study of random walks on these families were considered in [10]. In the present work, we illustrate that these random walks never exhibit the so-called product condition, and therefore also never display total variation cutoff as defined by Aldous and Diaconis [1]. In particular, we provide a large family of algebro-combinatorially motivated examples of collections of Markov chains which satisfy some well-known algebraic heuristics for cutoff, while not actually having the property.