  Discrete time crystals, which are phases of matter that break the discrete
time translational symmetry of a periodically driven system, have generated a
wave of recent interest and activity. In this work, we propose a classical
system of weakly nonlinear parametrically driven coupled oscillators as a
testbed to understand these phases. Such a system of oscillators in the
presence of dissipation has a period-doubling instability only at long
wavelengths where a large number of oscillators are in phase. To show that this
instability leads to a discrete time crystal when exposed to noise and
non-linearity, we derive a general condition to predict when a Markov process
will break time-translation symmetry. The first requirement of this condition
is that the Markov chain over doubled time steps has a stationary probability
distribution that spontaneously breaks a local symmetry. The second requirement
is that the order parameter flips sign over a single time step. We show that
our system of oscillators features period-doubling and yields a stationary
probability distribution given by a spatially continuous Ginzburg-Landau
Hamiltonian with spontaneous symmetry breaking. We then discuss applications of
the general condition to existing time crystal platforms.

Statistical Mechanics;Quantum Gases