  We consider the number distribution of topological defects resulting from the
finite-time crossing of a continuous phase transition and identify signatures
of universality beyond the mean value, predicted by the Kibble-Zurek mechanism.
Statistics of defects follows a binomial distribution with $\mathcal{N}$
Bernouilli trials associated with the probability of forming a topological
defect at the locations where multiple domains merge. All cumulants of the
distribution are predicted to exhibit a common universal power-law scaling with
the quench time in which the transition is crossed. Knowledge of the
distribution is used to discuss the onset of adiabatic dynamics and bound rare
events associated with large deviations.
