  We revisit the nonlinear stability of the critical invasion front in the
Ginzburg-Landau equation. Our main result shows that the amplitude of localized
perturbations decays with rate $t^{-3/2}$, while the phase decays diffusively.
We thereby refine earlier work of Bricmont and Kupiainen as well as Eckmann and
Wayne, who separately established nonlinear stability but with slower decay
rates. On a technical level, we rely on sharp linear estimates obtained through
analysis of the resolvent near the essential spectrum via a far-field/core
decomposition which is well suited to accurately describing the dynamics of
separate neutrally stable modes arising from far-field behavior on the left and
right.
