In equilibrium, a first-order magnetic transition is abrupt: the magnetization jumps. In the spatial picture, the minority phase nucleates as droplets, the droplets grow, and eventually one phase percolates — forms a connected cluster spanning the system. At equilibrium, the percolation transition coincides with the thermodynamic transition. They're the same event.
Ciliberto et al. (arXiv:2603.13127) quench a system across a first-order magnetic transition and find that the percolation transition occurs at a finite critical time after the quench — not instantaneously, and not at the same moment as the thermodynamic transition. The thermodynamic order parameter changes first. The geometric connectivity catches up later.
The mechanism: after a sudden quench, the new equilibrium phase appears as small, disconnected domains scattered across the system. The domains grow individually, and their growth rate depends on local conditions — curvature, neighboring domains, fluctuations. Percolation requires a connected path across the system, which needs the domains to grow large enough to touch. This takes time even after the average composition has shifted to favor the new phase.
The delay is measurable and depends on quench depth. A deeper quench (farther from the transition) produces more numerous but smaller nucleation centers, which take longer to coalesce. A shallower quench produces fewer but larger domains that connect faster. The percolation time is non-monotonic: there's an optimal quench depth that minimizes the time to connectivity.
This matters for any system where bulk properties depend on connectivity rather than composition — electrical conductivity in metal-insulator composites, fluid permeability in porous media, signal propagation in neural networks. In these systems, the relevant transition isn't when the new phase becomes thermodynamically dominant but when it becomes geometrically connected. The delay between composition and connectivity is the window where bulk properties lag their equilibrium values.