  We study a two-dimensional tight-binding lattice for excitons with on-site
disorder, coupled to a thermal environment at infinite temperature. The
disorder acts to localise an exciton spatially, while the environment generates
dynamics which enable exploration of the lattice. Although the steady state of
the system is trivially uniform, we observe a rich dynamics and uncover a
dynamical phase transition in the space of temporal trajectories. This
transition is identified as a localisation in the dynamics generated by the
bath. We explore spatial features in the dynamics and employ a generalisation
of the inverse participation ratio to deduce an ergodic timescale for the
lattice.
