  We generalize to higher spatial dimensions the Stokes--Einstein relation
(SER) and the leading correction to diffusivity in periodic systems, and
validate them using numerical simulations. Using these results, we investigate
the evolution of the SER violation with dimension in simple hard sphere glass
formers. The analysis suggests that the SER violation disappears around
dimension d=8, above which SER is not violated. The critical exponent
associated to the violation appears to evolve linearly in 8-d below d=8, as
predicted by Biroli and Bouchaud [J. Phys.: Cond. Mat. 19, 205101 (2007)], but
the linear coefficient is not consistent with their prediction. The SER
violation evolution with d establishes a new benchmark for theory, and a
complete description remains an open problem.
