  The superfluid phase transition dynamics and associated spontaneous vortex
formation with the crossing of the critical temperature in a disk geometry is
studied in the framework of the $AdS/CFT$ correspondence by solving the
Einstein-Abelian-Higgs model in an $AdS_4$ black hole. For a slow quench, the
vortex density admits a universal scaling law with the cooling rate as
predicted by the Kibble-Zurek mechanism (KZM), while for fast quenches, the
density shows a universal scaling behavior as a function of the final
temperature, that lies beyond the KZM prediction. The vortex number
distribution in both the power-law and saturation regimes can be approximated
by a normal distribution. However, the study of the universal scaling of the
cumulants reveals non-normal features and indicates that vortex statistics in
the newborn superfluid is best described by the Poisson binomial distribution,
previously predicted in the KZM regime [Phys. Rev. Lett. 124, 240602 (2020)].
This is confirmed by studying the cumulant scalings as a function of the quench
time and the quench depth. Our work supports the existence of a universal
defect number distribution that accommodates the KZM scaling, its breakdown at
fast quenches, and the additional universal scaling laws as a function of the
final value of the control parameter.
