  We use a probabilistic method to compute the propagation of an ionization
front corresponding to the re-ionization of the intergalactic medium in a LCDM
cosmology, including both hydrogen and helium. The effects of radiative
transfer substantially boost the temperature of the ionized gas over the case
of uniform re-ionization. The resulting temperature-density relation of the
ionized gas is both non-monotonic and multiple-valued, reflecting the non-local
character of radiative transfer and suggesting that a single polytropic
relation between local gas density and temperatue is a poor description of the
thermodynamic state of baryons in the post-reionization universe.
