  An unpolarized charge current passing through a chaotic quantum dot with
spin-orbit coupling can produce a spin polarized exit current without magnetic
fields or ferromagnets. We use random matrix theory to estimate the typical
spin polarization as a function of the number of channels in each lead in the
limit of large spin-orbit coupling. We find rms spin polarizations up to 45%
with one input channel and two output channels. Finite temperature and
dephasing both suppress the effect, and we include dephasing effects using a
new variation of the third lead model. If there is only one channel in the
output lead, no spin polarization can be produced, but we show that dephasing
lifts this restriction.
