  We investigate the lattice spacing dependence of the equilibration time for a
recently proposed multiscale thermalization algorithm for Markov chain Monte
Carlo simulations. The algorithm uses a renormalization-group matched coarse
lattice action and prolongation operation to rapidly thermalize decorrelated
initial configurations for evolution using a corresponding target lattice
action defined at a finer scale. Focusing on non-topological long-distance
observables in pure SU(3) gauge theory, we provide quantitative evidence that
the slow modes of the Markov process, which provide the dominant contribution
to the rethermalization time, have a suppressed contribution toward the
continuum limit, despite their associated timescales increasing. Based on these
numerical investigations, we conjecture that the prolongation operation used
herein will produce ensembles that are indistinguishable from the target
fine-action distribution for a sufficiently fine coupling at a given level of
statistical precision, thereby eliminating the cost of rethermalization.
