  Convergence with respect to imaginary-time discretization is an essential
part of any path-integral-based calculation. However, an unfortunate property
of existing non-preconditioned numerical integration schemes for path-integral
molecular dynamics (PIMD) - including ring-polymer molecular dynamics (RPMD)
and thermostatted RPMD (T-RPMD) - is that for a given MD timestep, the overlap
between the exact ring-polymer Boltzmann-Gibbs distribution and that sampled
using MD becomes zero in the infinite-bead limit. This has clear implications
for hybrid Metropolis Monte-Carlo/MD sampling schemes. We show that these
problems can be avoided through the introduction of "dimension-free" numerical
integration schemes for which the sampled ring-polymer position distribution
has non-zero overlap with the exact distribution in the infinite-bead limit for
the case of a harmonic potential. We show that dimension freedom can be
achieved via mollification of the forces from the physical potential and with
the BCOCB integration scheme. The dimension-free numerical integration schemes
yield finite error bounds for a given MD timestep as the number of beads is
taken to infinity; these conclusions are proven for harmonic potential and
borne out numerically for anharmonic systems, including water. The numerical
results for BCOCB are particularly striking, allowing for three-fold increases
in the stable timestep for liquid water with respect to the Bussi-Parrinello
(OBABO) and Leimkuhler (BAOAB) integrators while introducing negligible errors
in the statistical properties and absorption spectrum. Importantly, the
dimension-free, non-preconditioned integration schemes introduced here preserve
ergodicity and global second-order accuracy, and they remain simple, black-box
methods that avoid additional computational costs, tunable parameters, or
system-specific implementations.
