  When a quantum phase transition is crossed in finite time, critical slowing
down leads to the breakdown of adiabatic dynamics and the formation of
topological defects. The average density of defects scales with the quench rate
following a universal power-law predicted by the Kibble-Zurek mechanism. We
analyze the full counting statistics of kinks and report the exact kink number
distribution in the transverse-field quantum Ising model. Kink statistics is
described by the Poisson binomial distribution with all cumulants exhibiting a
universal power-law scaling with the quench rate. In the absence of finite-size
effects, the distribution approaches a normal one, a feature that is expected
to apply broadly in systems described by the Kibble-Zurek mechanism.
