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We analyse biased ensembles of trajectories for the random-field Ising modelon a fully-connected lattice, which is described exactly by mean-field theory.By coupling the activity of the system to a dynamical biasing field, we find arange of dynamical phase transitions, including spontaneous symmetry breakinginto ordered states. For weak bias, the phase behaviour is controlled byextrema of the free energy, which may be local minima or saddle points. Forlarge bias, the system tends to states of extremal activity, which may differstrongly from free energy minima. We discuss connections of these results torandom first-order transition theory of glasses, which motivates an extensionof the analysis to random-field Ising models where the dynamical activity isnot symmetric under magnetisation reversal.

Dynamical phase transition in the activity-biased fully-connected random field Ising model: connection with glass-forming systems