Relation between structure of blocked clusters and relaxation dynamics in kinetically constrained models

We investigate the relation between the cooperative length and 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, τp, which is the time until a particle moves for the first time, and the culling time, τc, which is the minimal number of particles that need to move before a specific particle can move, τp=τcγ, 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/τβ, but unlike previous works, we find that the exponent β appears to decay to 0 as the particle density approaches 1.