  We present a study of the 3d O(2) non-linear $\sigma$-model on the lattice,
which exhibits topological defects in the form of vortices. They tend to
organize into vortex lines that bear close analogies with global cosmic
strings. Therefore, this model serves as a testbed for studying the dynamics of
topological defects. It undergoes a second order phase transition, hence it is
appropriate for investigating the Kibble-Zurek mechanism. In this regard, we
explore the persistence of topological defects when the temperature is rapidly
reduced from above to below the critical temperature; this cooling (or
"quenching") process takes the system out of equilibrium. We probe a wide range
of inverse cooling rates $\tau_{\rm Q}$ and final temperatures, employing
distinct Monte Carlo algorithms. The results consistently show that the density
of persisting topological defects follows a power-law in $\tau_{\rm Q}$, in
agreement with Zurek's conjecture. On the other hand, at this point our results
do not confirm Zurek's prediction for the exponent in this power-law, but its
final test is still under investigation.
