  The zero-temperature Glauber dynamics of the random-field Ising model
describes various ubiquitous phenomena such as avalanches, hysteresis, and
related critical phenomena. Here, for a model on a random graph with a special
initial condition, we derive exactly an evolution equation for an order
parameter. Through a bifurcation analysis of the obtained equation, we reveal a
new class of cooperative slow dynamics with the determination of critical
exponents.
