Is there a fractional breakdown of the Stokes-Einstein relation in kinetically constrained models at low temperature?

We study the motion of a tracer particle injected in facilitated models which are used to model supercooled liquids in the vicinity of the glass transition. We consider the East model, FA1f model and a more general class of non-cooperative models. For East previous works had identified a fractional violation of the Stokes-Einstein relation with a decoupling between diffusion and viscosity of the form D similar to tau(-xi) with xi similar to 0.73. We present rigorous results proving that instead log(D) = - log(tau) + O(log(1/q)), which implies at leading order log(D)/log(tau) similar to -1 for very large time scales. Our results do not exclude the possibility of SE breakdown, albeit non-fractional. Indeed extended numerical simulations by other authors show the occurrence of this violation and our result suggests D tau similar to 1/q(alpha), where q is the density of excitations. For FA1f we prove a fractional Stokes-Einstein relation in dimension 1, and D similar to tau(-1) in dimension 2 and higher, confirming previous works. Our results extend to a larger class of non-cooperative models. Copyright (C) EPLA, 2014