Scaling properties of multiscale equilibration

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 nontopological 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.

Figure 1

Rethermalization time dependence of the Yang-Mills action density at flow time $w_{0.4}^2/4$ for $a=0.06\mathrm{fm}$, $a=0.04\mathrm{fm}$, $a=0.03\mathrm{fm}$, and $a=0.02\mathrm{fm}$. Note that each data point is statistically independent and the error bands are for uncorrelated exponential fits to the data (${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ range: 0.6–2.1).

Figure 2

Lattice spacing dependence of the rethermalization timescale (${\tau }_R$) and coupling ($c_R$), determined from single-exponential fits to the rethermalization time dependence of the action density at flow time $w_{0.4}^2/4$. The solid line and error band indicate a linear fit to all available data points.