  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.
Furthermore, KCM have an interest in their own since they display some of the
most striking features of the liquid-glass transition, a major and longstanding
open problem in condensed matter physics. A key issue for KCM is to identify
the scaling of the characteristic time scales when the equilibrium density of
empty sites, $q$, goes to zero. In [19,20] a general scheme was devised to
determine a sharp upper bound for these time scales. Our paper is devoted to
developing a (very different) technique which allows to prove matching lower
bounds. We analyse the class of two-dimensional supercritical rooted KCM and
the Duarte KCM, the most studied critical $1$-rooted model. We prove that the
relaxation time and the mean infection time diverge for supercritical rooted
KCM as $e^{\Theta((\log q)^2)}$ and for Duarte KCM as $e^{\Theta((\log
q)^4/q^2)}$ when $q\downarrow 0$. These results prove the conjectures put
forward in [20,22], and establish that the time scales for these KCM diverge
much faster than for the corresponding $\mathcal U$-bootstrap processes, the
main reason being the occurrence of energy barriers which determine the
dominant behaviour for KCM, but which do not matter for the bootstrap dynamics.
