  We calculate the corrections to the thermodynamic limit of the critical
density for jamming in the Kob-Andersen and Fredrickson-Andersen
kinetically-constrained models, and find them to be finite-density corrections,
and not finite-size corrections. We do this by introducing a new numerical
algorithm, which requires negligible computer memory since contrary to
alternative approaches, it generates at each point only the necessary data. The
algorithm starts from a single unfrozen site and at each step randomly
generates the neighbors of the unfrozen region and checks whether they are
frozen or not. Our results correspond to systems of size greater than
10^7x10^7, much larger than any simulated before, and are consistent with the
rigorous bounds on the asymptotic corrections. We also find that the average
number of sites that seed a critical droplet is greater than 1.
