  Current fluctuations in boundary-driven diffusive systems are, in many cases,
studied using hydrodynamic theories. Their predictions are then expected to be
valid for currents which scale inversely with the system size. To study this
question in detail, we introduce a class of large-N models of one-dimensional
boundary-driven diffusive systems, whose current large deviation functions are
exactly derivable for any finite number of sites. Surprisingly, we find that
for some systems the predictions of the hydrodynamic theory may hold well
beyond their naive regime of validity. Specifically, we show that, while a
symmetric partial exclusion process exhibits non-hydrodynamic behaviors
sufficiently far beyond the naive hydrodynamic regime, a symmetric inclusion
process is well described by the hydrodynamic theory for arbitrarily large
currents. We conjecture, and verify for zero-range processes, that the
hydrodynamic theory captures the statistics of arbitrarily large currents for
all models where the mobility coefficient as a function of density is unbounded
from above. In addition, for the large-N models, we prove the additivity
principle under the assumption that the large deviation function has no
discontinuous transitions.
