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We analyze the density and size dependence of the relaxation time $\tau$ forkinetically constrained spin systems. These have been proposed as models forstrong or fragile glasses and for systems undergoing jamming transitions. Forthe one (FA1f) or two (FA2f) spin facilitated Fredrickson-Andersen model at anydensity $\rho<1$ and for the Knight model below the critical density at whichthe glass transition occurs, we show that the persistence and the spin-spintime auto-correlation functions decay exponentially. This excludes thestretched exponential relaxation which was derived by numerical simulations.For FA2f in $d\geq 2$, we also prove a super-Arrhenius scaling of the form$\exp(1/(1-\rho))\leq \tau\leq\exp(1/(1-\rho)^2)$. For FA1f in $d$=$1,2$ werigorously prove the power law scalings recently derived in \cite{JMS} while in$d\geq 3$ we obtain upper and lower bounds consistent with findings therein.Our results are based on a novel multi-scale approach which allows to analyze$\tau$ in presence of kinetic constraints and to connect time-scales anddynamical heterogeneities. The techniques are flexible enough to allow avariety of constraints and can also be applied to conservative stochasticlattice gases in presence of kinetic constraints.Comment: 4 page

Relaxation times of kinetically constrained spin models with glassy dynamics