Spectral condition for biased card shuffling

Establish whether the spectral condition fref{cond} is satisfied for the Metropolis biased card-shuffling chain, also known as the multi-species ASEP on a finite interval or the random Metropolis scan.

Background

The Metropolis biased card-shuffling chain is a reversible card shuffle with Mallows stationary measure, in which adjacent cards are exchanged at rates depending on whether they are ordered. The paper cites known cutoff and limit-profile results for this model.

Although the limit profile is known and can be checked to be continuous, the paper identifies the corresponding f2 analysis near the cutoff as unresolved. In particular, it is not known whether the spectral bound imposed in condition \eqref{cond} holds for this chain.

References

The $\ell_2$ analysis is still open and condition cond is not known to be satisfied, yet knowing the limit profile formula, one can check that it is indeed continuous.

— Continuity for Limit Profiles of Reversible Markov Chains  (2503.09550 - Nestoridi, 12 Mar 2025) in Section 5, subsection “Biased Card Shuffling and ASEP”