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The dynamical behaviours of a kinetically constrained spin model(Fredrickson-Andersen model) on a Bethe lattice are investigated by aperturbation analysis that provides exact final states above the nonergodictransition point. It is observed that the time-dependent solutions of thederived dynamical systems obtained by the perturbation analysis becomesystematically closer to the results obtained by Monte Carlo simulations as theorder of a perturbation series is increased. This systematic perturbationanalysis also clarifies the existence of a dynamical scaling law, whichprovides a implication for a universal relation between a size scale and a timescale near the nonergodic transition.

Systematic perturbation approach for a dynamical scaling law in a kinetically constrained spin model