  The East model is the simplest one-dimensional kinetically-constrained model
of $N$ spins with a trivial equilibrium that displays anomalously large
spatio-temporal fluctuations, with characteristic "space-time bubbles" in
trajectory space, and with a discontinuity at the origin for the first
derivative of the scaled cumulant generating function of the total activity.
These striking dynamical properties are revisited via the large deviations at
various levels for the relevant local empirical densities and activities that
only involve two consecutive spins. This framework allows to characterize their
anomalous rate functions and to analyze the consequences for all the
time-additive observables that can be reconstructed from them, both for the
pure and for the random East model. These singularities in dynamical large
deviations properties disappear when the hard-constraint of the East model is
replaced by the soft constraint.
