  The additivity principle (AP) allows to compute the current distribution in
many one-dimensional (1d) nonequilibrium systems. Here we extend this
conjecture to general d-dimensional driven diffusive systems, and validate its
predictions against both numerical simulations of rare events and microscopic
exact calculations of three paradigmatic models of diffusive transport in d=2.
Crucially, the existence of a structured current vector field at the
fluctuating level, coupled to the local mobility, turns out to be essential to
understand current statistics in d>1. We prove that, when compared to the
straightforward extension of the AP to high-d, the so-called weak AP always
yields a better minimizer of the macroscopic fluctuation theory action for
current statistics.
