We analyze a Markov chain, known as the product replacement chain, on the set of generating n-tuples of a fixed finite group G. We show that as n→ ∞, the total-variation mixing time of the chain has a cutoff at time 32nlogn with window of order n. This generalizes a result of Ben-Hamou and Peres (who established the result for G= Z/ 2) and confirms a conjecture of Diaconis and Saloff-Coste that for an arbitrary but fixed finite group, the mixing time of the product replacement chain is O(nlog n).
- Cutoff phenomenon
- Markov chain
- Mixing time
- Product replacement algorithm
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty