  We study the connection between the phase behaviour of quantum dimers and the
dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK)
point a quantum dimer Hamiltonian is equivalent to the Markov generator of the
dynamics of classical dimers. A less well understood fact is that away from the
RK point the quantum-classical connection persists: in this case the
Hamiltonian corresponds to a non-stochastic "tilted" operator that encodes the
statistics of time-integrated observables of the classical stochastic problem.
This implies a direct relation between the phase behaviour of quantum dimers
and properties of ensembles of stochastic trajectories of classical dimers. We
make these ideas concrete by studying fully packed dimers on the square
lattice. Using transition path sampling - supplemented by trajectory umbrella
sampling - we obtain the large deviation statistics of dynamical activity in
the classical problem, and show the correspondence between the phase behaviour
of the classical and quantum systems. The transition at the RK point between
quantum phases of distinct order corresponds, in the classical case, to a
trajectory phase transition between active and inactive dynamical phases.
Furthermore, from the structure of stochastic trajectories in the active
dynamical phase we infer that the ground state of quantum dimers has columnar
order to one side of the RK point. We discuss how these results relate to those
from quantum Monte Carlo, and how our approach may generalise to other
problems.
