We present a detailed study of the equilibrium properties and stochastic dynamic evolution of the U(1)-invariant relativistic complex field theory in three dimensions. This model has been used to describe, in various limits, properties of relativistic bosons at finite chemical potential, type II su- perconductors, magnetic materials and aspects of cosmology. We characterise the thermodynamic second-order phase transition in different ways. We study the equilibrium vortex configurations and their statistical and geometrical properties in equilibrium at all temperatures. We show that at very high temperature the statistics of the filaments is the one of fully-packed loop models. We identify the temperature, within the ordered phase, at which the number density of vortex lengths falls-off algebraically and we associate it to a geometric percolation transition that we characterise in various ways. We measure the fractal properties of the vortex tangle at this threshold. Next, we perform infinite rate quenches from equilibrium in the disordered phase, across the thermo- dynamic critical point, and deep into the ordered phase. We show that three time regimes can be distinguished: a first approach towards a state that, within numerical accuracy, shares many features with the one at the percolation threshold, a later coarsening process that does not alter, at sufficiently low temperature, the fractal properties of the long vortex loops, and a final approach to equilibrium. These features are independent of the reconnection rule used to build the vortex lines. In each of these regimes we identify the various length-scales of the vortices in the system. We also study the scaling properties of the ordering process and the progressive annihilation of topological defects and we prove that the time-dependence of the time-evolving vortex tangle can be described within the dynamic scaling framework.
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