Improving coupling-based spectral gap bounds to avoid exponential dependence on dimension

Investigate whether the combination of geometric drift conditions with close-coupling (local Doeblin/minorization) derived from total-variation Hölder continuity of conditionals can be refined to yield spectral gap bounds for Gibbs samplers that do not deteriorate exponentially with the dimension d.

Background

Sequential coupling yields close-coupling conditions reminiscent of local Doeblin conditions; together with drift/minorization theory, this can provide geometric convergence rates.

A direct use of these conditions here gives spectral gap bounds that scale exponentially poorly with dimension, and it is presently unclear whether the approach can be sharpened.