  The East process is a 1D kinetically constrained interacting particle system,
introduced in the physics literature in the early 90's to model liquid-glass
transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that
its mixing time on $L$ sites has order $L$. We complement that result and show
cutoff with an $O(\sqrt{L})$-window.
  The main ingredient is an analysis of the front of the process (its rightmost
zero in the setup where zeros facilitate updates to their right). One expects
the front to advance as a biased random walk, whose normal fluctuations would
imply cutoff with an $O(\sqrt{L})$-window. The law of the process behind the
front plays a crucial role: Blondel showed that it converges to an invariant
measure $\nu$, on which very little is known. Here we obtain quantitative
bounds on the speed of convergence to $\nu$, finding that it is exponentially
fast. We then derive that the increments of the front behave as a stationary
mixing sequence of random variables, and a Stein-method based argument of
Bolthausen ('82) implies a CLT for the location of the front, yielding the
cutoff result.
  Finally, we supplement these results by a study of analogous kinetically
constrained models on trees, again establishing cutoff, yet this time with an
$O(1)$-window.
