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 $\tau_\mathrm{Q}$ and final temperatures, employing distinct Monte Carlo
algorithms. The results consistently show that the density of persisting topological defects follows
a power-law in $\tau_\mathrm{Q}$, 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.