Excessive symmetry can preclude cutoff

TL;DR

This paper shows that certain symmetric random walks on Kneser graphs do not exhibit the cutoff phenomenon, providing examples where algebraic heuristics for cutoff fail.

Contribution

The authors demonstrate that random walks on nested Kneser graphs with high symmetry do not satisfy the product condition, thus lacking cutoff, challenging common heuristics.

Findings

Random walks on Kneser graphs never satisfy the product condition.

These walks do not exhibit total variation cutoff.

High symmetry can prevent cutoff despite algebraic heuristics.

Abstract

For each $n,r \geq 0$, let $KG(n,r)$ denote the Kneser Graph; that whose vertices are labeled by $r$-element subsets of $n$, and whose edges indicate that the corresponding subsets are disjoint. Fixing $r$ and allowing $n$ to vary, one obtains a family of nested graphs, each equipped with a natural action by a symmetric group $\mathfrak{S}_n$, such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in [RW], while a systematic study of random walks on these families were considered in [RW2]. In this paper 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 [AD]. 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.