  The effective increase of the critical density associated with the
interaction of relativistically intense laser pulses with overcritical plasmas,
known as self-induced transparency, is revisited for the case of circular
polarization. A comparison of particle-in-cell simulations to the predictions
of a relativistic cold-fluid model for the transparency threshold demonstrates
that kinetic effects, such as electron heating, can lead to a substantial
increase of the effective critical density compared to cold-fluid theory. These
results are interpreted by a study of separatrices in the single-electron phase
space corresponding to dynamics in the stationary fields predicted by the
cold-fluid model. It is shown that perturbations due to electron heating
exceeding a certain finite threshold can force electrons to escape into the
vacuum, leading to laser pulse propagation. The modification of the
transparency threshold is linked to the temporal pulse profile, through its
effect on electron heating.
