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.
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