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 $D\sim\tau^{-\xi}$ with $\xi\sim 0.73$. We present rigorous results proving that instead $\log(D)=-\log(\tau)+O(\log(1/q))$, which implies at leading order $\log(D)/\log(\tau)\sim -1$ 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 $D\tau\sim 1/q^\alpha$, where $q$ is the density of excitations. For FA1f we prove fractional Stokes Einstein in dimension $1$, and $D\sim\tau^{-1}$ in dimension $2$ and higher, confirming previous works. Our results extend to a larger class of non-cooperative models.