  The reionization of intergalactic hydrogen has received intense theoretical
scrutiny over the past two decades. Here, we approach the process formally as a
percolation process and phase transition. Using semi-numeric simulations, we
demonstrate that an infinitely-large ionized region abruptly appears at an
ionized fraction of ~0.1 and quickly grows to encompass most of the ionized
gas: by an ionized fraction of 0.3, nearly ninety percent of the ionized
material is part of this region. Throughout most of reionization, nearly all of
the intergalactic medium is divided into just two regions, one ionized and one
neutral, and both infinite in extent. We also show that the discrete ionized
regions that exist before and near this transition point follow a near-power
law distribution in volume, with equal contributions to the total filling
factor per logarithmic interval in size up to a sharp cutoff in volume. These
qualities are generic to percolation processes, with the detailed behavior a
result of long-range correlations in the underlying density field. These
insights will be crucial to understanding the distribution of ionized and
neutral gas during reionization and provide precise meaning to the intuitive
description of reionization as an "overlap" process.
