Equivalence of cutoff and the product condition for ‘reasonable’ models

Prove that, for an appropriately formalized class of ‘reasonable’ models of finite Markov chains, exhibiting total-variation cutoff is equivalent to satisfying the product condition lim_{n→∞} λ_n t_mix^{(n)} = ∞, where λ_n is the spectral gap and t_mix^{(n)} is the mixing time.

Background

The product condition (Peres, 2004) is necessary for cutoff but not sufficient in full generality; explicit counterexamples can be built via rank-one perturbations. Nonetheless, it is widely believed to predict cutoff on ‘reasonable’ families (e.g., sparse, structured chains). The text highlights the challenge of formalizing ‘reasonable’ and proving equivalence in that class, beyond special cases like birth-and-death chains and weighted trees.