Polynomial-time mixing for a matroid flag down-up walk

Prove that, for a matroid M and the probability distribution on maximal flags of flats defined by μ(F_1\subsetneq\dots\subsetneq F_d) proportional to \prod_{i=1}^d (|F_{i+1}|-|F_i|)/|F_{i+1}|, the associated down-up Markov chain mixes in time polynomial in d.

Background

The paper considers the path complex whose faces are flags of flats of a matroid and assigns each maximal flag a nonuniform weight proportional to the product of successive rank-size increments divided by the size of the upper flat. The associated down-up walk is a natural Markov chain for sampling such flags.

The authors note that the partition function of this distribution equals the last coefficient of the matroid’s characteristic polynomial. Consequently, a polynomial mixing-time guarantee would yield algorithmic consequences for counting and sampling problems that remain open in the cited literature.