  A general matrix-based scheme for analyzing the long-time dynamics in
kinetically constrained models such as the East model is presented. The
treatment developed here is motivated by the expectation that slowly-relaxing
spin domains of arbitrary size govern the highly cooperative events that lead
to spin relaxation at long times. To account for the role of large spin domains
in the dynamics, a complete basis expressed in terms of domains of all sizes is
introduced. It is first demonstrated that accounting for single domains of all
possible sizes leads to a simple analytical result for the two-time single-spin
correlation function in the East model that is in excellent quantitative
agreement with simulation data for equilibrium spin up density values c greater
or equal to 0.6. It is then shown that including also two neighboring domains
leads to a closed expression that describes the slow relaxation of the system
down to c approximately 0.3. Ingredients of generalizing the method to lower
values of c are also provided, as well as to other models. The main advantage
of this approach is that it gives explicit analytical results and that it
requires neither an arbitrary closure for the memory kernel nor the
construction of an irreducible memory kernel. It also allows one to calculate
quantities that measure heterogeneity in the same framework, as is illustrated
on the neighbor-pair correlation function and the distribution of relaxation
times.
