  We show, using the macroscopic fluctuation theory of Bertini, De Sole,
Gabrielli, Jona-Lasinio, and Landim, that the statistics of the current of the
symmetric simple exclusion process (SSEP) connected to two reservoirs are the
same on an arbitrary large finite domain in dimension $d$ as in the one
dimensional case. Numerical results on squares support this claim while results
on cubes exhibit some discrepancy. We argue that the results of the macroscopic
fluctuation theory should be recovered by increasing the size of the contacts.
The generalization to other diffusive systems is straightforward.
