  We use projected entangled-pair states (PEPS) to calculate the large
deviations (LD) statistics of the dynamical activity of the two dimensional
East model, and the two dimensional symmetric simple exclusion process (SSEP)
with open boundaries, in lattices of up to 40x40 sites. We show that at
long-times both models have phase transitions between active and inactive
dynamical phases. For the 2D East model we find that this trajectory transition
is of the first-order, while for the SSEP we find indications of a second order
transition. We then show how the PEPS can be used to implement a trajectory
sampling scheme capable of directly accessing rare trajectories. We also
discuss how the methods described here can be extended to study rare events at
finite times.
