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We describe the motion of a tracer in an environment given by a kineticallyconstrained spin model (KCSM) at equilibrium. We check convergence of itstrajectory properly rescaled to a Brownian motion and positivity of thediffusion coefficient $D$ as soon as the spectral gap of the environment ispositive (which coincides with the ergodicity region under general conditions).Then we study the asymptotic behavior of $D$ when the density $1-q$ of theenvironment goes to $1$ in two classes of KCSM. For noncooperative models, thediffusion coefficient $D$ scales like a power of $q$, with an exponent that wecompute explicitly. In the case of the Fredrickson-Andersen one-spinfacilitated model, this proves a prediction made in Jung, Garrahan and Chandler[Phys. Rev. E 69 (2004) 061205]. For the East model, instead we prove that thediffusion coefficient is comparable to the spectral gap, which goes to zerofaster than any power of $q$. This result contradicts the prediction ofphysicists (Jung, Garrahan and Chandler [Phys. Rev. E 69 (2004) 061205; J.Chem. Phys. 123 (2005) 084509]), based on numerical simulations, that suggested$D\sim \operatorname {gap}^{\xi}$ with $\xi<1$.

Tracer diffusion at low temperature in kinetically constrained models