  We consider the equilibrium dynamics of Ising spin models with multi-spin
interactions on sparse random graphs (Bethe lattices). Such models undergo a
mean field glass transition upon increasing the graph connectivity or lowering
the temperature. Focusing on the low temperature limit, we identify the large
scale rearrangements responsible for the dynamical slowing-down near the
transition. We are able to characterize exactly the dynamics near criticality
by analyzing the statistical properties of such rearrangements. Our approach
can be generalized to a large variety of glassy models on sparse random graphs,
ranging from satisfiability to kinetically constrained models.
