Vortex formation in two dimensions: When symmetry breaks, how big are the pieces?

We investigate the dynamics of second order phase transitions in two dimensions, breaking a gauged U(1) symmetry. Using numerical simulations, we show that the density of topological defects formed scales with the quench timescale ${\tau }_Q$ as $n\sim {\tau }_Q^{-1\mathrm{/}2}$ when the dynamics is overdamped at the instant when the freezeout of thermal fluctuations takes place, and $n \sim \tau_Q^{-2/3}$ in the underdamped case. This is predicted by the scenario proposed by one of us [1].