Dynamical random field Ising model at zero temperature

In this paper, we study the evolution of the zero-temperature random field Ising model as the mean of the external field $M$ increases from $-\mathrm{\infty }$ to $\mathrm{\infty }$. We focus on two types of evolutions: the ground state evolution and the Glauber evolution. For the ground state evolution, we investigate the occurrence of global avalanche, a moment where a large fraction of spins flip simultaneously from minus to plus. In two dimensions, no global avalanche occurs, while in three or higher dimensions, there is a phase transition: a global avalanche happens when the noise intensity is small, but not when it is large. Additionally, we study the zero-temperature Glauber evolution, where spins are updated locally to minimize the Hamiltonian. Our results show that for small noise intensity, in dimensions $d=2$ or $3$, most spins flip around a critical time $c_d=\frac{2\sqrt{d}}{1+\sqrt{d}}$ (but we cannot decide whether such flipping occurs simultaneously or not). We also connect this process to polluted bootstrap percolation and solve an open problem on it.