A variation of strong stationary times for random walks with partial symmetries

TL;DR

This paper proposes a new approach to analyze the mixing times of random walks on the symmetric group by focusing on interactions between pairs of cards, offering potentially tighter bounds.

Contribution

It introduces a variation of strong stationary times based on pair interactions, differing from traditional methods that accumulate larger blocks of cards.

Findings

Provides a new method for bounding mixing times using pair interactions.

Offers potential improvements over traditional block-based approaches.

Applicable to random walks with partial symmetries.

Abstract

We introduce a variation of strong stationary times for random walks on the symmetric group. Rather than proceed in the usual fashion of accumulating larger and larger blocks of cards which may be in any order, we wait for pairs of cards to `interact', and bound the mixing time by the time taken for enough interactions to occur.