  Kinetically constrained models (KCM) are reversible interacting particle
systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type,
which represent a natural stochastic (and non-monotone) counterpart of the
family of cellular automata known as $\mathcal U$-bootstrap percolation. KCM
also display some of the peculiar features of the so-called "glassy dynamics",
and as such they are extensively used in the physics literature to model the
liquid-glass transition, a major and longstanding open problem in condensed
matter physics.
  We consider two-dimensional KCM with update rule $\mathcal U$, and focus on
proving universality results for the mean infection time of the origin, in the
same spirit as those recently established in the setting of $\mathcal
U$-bootstrap percolation. We first identify what we believe are the correct
universality classes, which turn out to be different from those of $\mathcal
U$-bootstrap percolation. We then prove universal upper bounds on the mean
infection time within each class, which we conjecture to be sharp up to
logarithmic corrections. In certain cases, including all supercritical models,
and the well-known Duarte model, our conjecture has recently been confirmed in
[MMT]. In fact, in these cases our upper bound is sharp up to a constant factor
in the exponent. For certain classes of update rules, it turns out that the
infection time of the KCM diverges much faster than for the corresponding
$\mathcal U$-bootstrap process when the equilibrium density of infected sites
goes to zero. This is due to the occurrence of energy barriers which determine
the dominant behaviour for KCM, but which do not matter for the monotone
bootstrap dynamics.
