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The relation between the structure of blocked clusters and the relaxation dynamics in kinetically-constrained models

We investigate the relation between the cooperative length and the relaxation time, represented respectively by the culling time and the persistence time, in the Fredrickson-Andersen, Kob-Andersen and spiral kinetically-constrained models. By mapping the dynamics to diffusion of defects, we find a relation between the persistence time, $τ_{p}$, which is the time until a particle moves for the first time, and the culling time, $τ_{c}$, which is the minimal number of particles that need to move before a specific particle can move, $τ_{p}=τ^γ_{c}$, where $γ$ is model- and dimension dependent. We also show that the persistence function in the Kob-Andersen and Fredrickson-Andersen models decays subexponentially in time, $P(t)=exp[-(t/τ)^β]$, but unlike previous works we find that the exponent $β$ appears to decay to 0 as the particle density approaches 1.