  Here we demonstrate that tensor network techniques - originally devised for
the analysis of quantum many-body problems - are well suited for the detailed
study of rare event statistics in kinetically constrained models (KCMs). As
concrete examples we consider the Fredrickson-Andersen and East models, two
paradigmatic KCMs relevant to the modelling of glasses. We show how variational
matrix product states allow to numerically approximate - systematically and
with high accuracy - the leading eigenstates of the tilted dynamical generators
which encode the large deviation statistics of the dynamics. Via this approach
we can study system sizes beyond what is possible with other methods, allowing
us to characterise in detail the finite size scaling of the trajectory-space
phase transition of these models, the behaviour of spectral gaps, and the
spatial structure and "entanglement" properties of dynamical phases. We discuss
the broader implications of our results.
