  The study of dynamical large deviations allows for a characterization of
stationary states of lattice gas models out of equilibrium conditioned on
averages of dynamical observables. The application of this framework to the
two-dimensional random walk conditioned on partial currents reveals the
existence of a dynamical phase transition between delocalized band dynamics and
localized vortex dynamics. We present a numerical microscopic characterization
of the phases involved, and provide analytical insight based on the macroscopic
fluctuation theory. A spectral analysis of the microscopic generator shows that
the continuous phase transition is accompanied by spontaneous
$\mathbb{Z}_2$-symmetry breaking whereby the stationary solution loses the
reflection symmetry of the generator. Dynamical phase transitions similar to
this one, which do not rely on exclusion effects or interactions, are likely to
be observed in more complex non-equilibrium physics models.
