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The random-field Ising model shows extreme critical slowdown that has beendescribed by activated dynamic scaling: the characteristic time for therelaxation to equilibrium diverges exponentially with the correlation length,$\ln \tau\sim \xi^\psi/T$ , with $\psi$ an \textit{a priori} unknown barrierexponent. Through a nonperturbative functional renormalization group, we showthat 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 thatcontrols 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 theupper critical dimension $d=6$, $\lambda=1$ so that $\psi=0$, and activatedscaling gives way to conventional scaling. We give a physical interpretation ofthe results in terms of collective events in real space, avalanches anddroplets. We also propose a way to check the two regimes by computersimulations of long-range 1-$d$ systems.

Activated dynamic scaling in the random-field Ising model: a nonperturbative functional renormalization group approach