  The random-field Ising model shows extreme critical slowdown that has been
described by activated dynamic scaling: the characteristic time for the
relaxation to equilibrium diverges exponentially with the correlation length,
$\ln \tau\sim \xi^\psi/T$ , with $\psi$ an \textit{a priori} unknown barrier
exponent. Through a nonperturbative functional renormalization group, we show
that for spatial dimensions $d$ less than a critical value $d_{DR} \simeq 5.1$,
also associated with dimensional-reduction breakdown, $\psi=\theta$ with
$\theta$ the temperature exponent near the zero-temperature fixed point that
controls the critical behavior. For $d>d_{DR}$ on the other hand,
$\psi=\theta-2\lambda$ where $\theta=2$ and $\lambda>0$ a new exponent. At the
upper critical dimension $d=6$, $\lambda=1$ so that $\psi=0$, and activated
scaling gives way to conventional scaling. We give a physical interpretation of
the results in terms of collective events in real space, avalanches and
droplets. We also propose a way to check the two regimes by computer
simulations of long-range 1-$d$ systems.
