  We study the convergence properties of Glauber dynamics for the random field
Ising model (RFIM) with ferromagnetic interactions on finite domains of
$\mathbb{Z}^d$, $d \ge 2$. Of particular interest is the Griffiths phase where
correlations decay exponentially fast in expectation over the quenched
disorder, but there exist arbitrarily large islands of weak fields where
low-temperature behavior is observed. Our results are twofold:
  1. Under weak spatial mixing (boundary-to-bulk exponential decay of
correlations) in expectation, we show that the dynamics satisfy a weak
Poincar\'e inequality -- equivalent to large-set expansion -- implying
algebraic relaxation to equilibrium over timescales polynomial in the volume
$N$ of the domain, and polynomial time mixing from a warm start. From this we
construct a polynomial-time approximate sampling algorithm based on running
Glauber dynamics over an increasing sequence of approximations of the domain.
  2. Under strong spatial mixing (exponential decay of correlations even near
boundary pinnings) in expectation, we prove a full Poincar\'e inequality,
implying exponential relaxation to equilibrium and $N^{o(1)}$-mixing time. Note
by way of example, both weak and strong spatial mixing hold at any temperature,
provided the external fields are strong enough.
  Our proofs combine a stochastic localization technique which has the effect
of increasing the variance of the field, with a field-dependent coarse graining
which controls the resulting sub-critical percolation process of sites with
weak fields.
